Dharma is eternal and fixed. Each varna in Hinduism has its own dharma, which is defined by the Vedas.
What is the dharma for Mlecchas?
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In this chapter of the Shanti Parva the solar dynasty king Mandhata asks Vishnu, who has taken the form of Indra, what Dharmas kings should force Mlecchas like Yavanas, Sakas, etc. to perform. Here is how Vishnu answers it:
Mandhatri said, ‘What duties should be performed by the Yavanas, the Kiratas, the Gandharvas, the Chinas, the Savaras, the Barbaras, the Sakas, the Tusharas, the Kankas, the Pathavas, the Andhras, the Madrakas, the Paundras, the Pulindas, the Ramathas, the Kamvojas, the several castes that have sprung up from Brahmanas and Kshatriyas, the Vaisyas, and the Sudras, that reside in the dominions of (Arya) kings? What are those duties again to the observance of which kings like ourselves should force those tribes that subsist by robbery? I desire to hear all this. O illustrious god, instruct me. O chief of all the deities, thou art the friend of us Kshatriyas.'
Indra said, 'All the robber tribes should serve their mothers and fathers, their preceptors and other seniors, and recluses living in the woods. All the robber tribes should also serve their kings. The duties and rites inculcated in the Vedas should also be followed by them. They should perform sacrifices in honour of the Pitris, dig wells, (and dedicate them to universal service), give water to thirsty travellers, give away beds and make other seasonable presents unto Brahmanas. Abstention from injury, truth, suppression of wrath, supporting Brahmanas and kinsmen by giving them their dues, maintenance of wives and children, purity, peacefulness, making presents to Brahmanas at sacrifices of every kind, are duties that should be practised by every person of this class who desire his own prosperity. Such a person should also perform all kinds of Paka-yajnas with costly presents of food and wealth. These and similar duties, O sinless one, were laid down in olden days for persons of this class. All these acts which have been laid down for all others should be done by persons of also the robber class, O king.'
Who are these Mlecchas to start with? Those who do not follow the rules of Hinduism.
Then why would Hindu scriptures delineate duties for them?
The following passage describes how things will be like in the Kali Yuga.
Almost all will be deceitful, cruel, vain, egoistic, thievish and mischievous. Men will be at variance with one another; women will be at variance with one another; no fear will exist in marriage ties. Properties will be only of those that will make them (i. e. there will cease to be any inheritance from father to son and so forth ). Husbands will be obedient to their wives; unchaste women will be in every house. Wives will rebuke their husbands by incessant noisings and chidings. Wives will be the sole mistresses of houses and husbands will stand before them as servants with folded palms. Fathers-in-law and mothers-in-law will be their servants. The brothers of wives, and their friends will be the managers of the household affairs. But there will be no friendship with one's own class fellows. The brothers and friends of the house owners (masters of the house) will appear quite strangers as if they are new-comers. Without the command of the house-wives, the masters of the houses will be unable to do anything. The divisions of caste (Brâhmanâ, Ksattriya, Vaisya, and S'ûdra) will entirely disappear. Far from practising Sandhyâ Bandanam and other daily practices, the Brâhmanas will cease to hold the holy threads even on their bodies. The four colour-classes will practise the doings of the Mlechchas, read the S'âstras of the Mlechchas and forsake their own S'âstras. The Brâhmins, Ksattriyas, and Vais'yas will become the slaves of S'ûdras, will become their cooks, runners and carriers of buffaloes. Every one will be devoid of truth.
Devi BhAgavata PurAna-Book 9-Chapter 8.
More from the same chapter:
The unchaste women who deal in giving loans, etc., and live on interests and the procuresses during menstruation will cook in Brâhmin families. There will be no distinction of food, no distinction of wombs, no distinction of Âs'ramas, and no distinction of persons. All will turn out Mlechchas. O Nârada! Thus, when the Kali will have its full play, the whole world will be filled with Mlechchas,
Then who are the Mlecchas as per these passages? Those who do not follow the Hindu Dharma at all.
Note that as per the first quote, the Mlecchas have their own Shastras and which are different from those of Hinduism. So, as per Hinduism they are required to follow their own Shastras, whichever those are.
Then, why would Hindu scriptures prescribe rules for them given that they are specifically described as the ones who do not follow those laws at all?
Now, if you are looking for verses that say - "righteousness is the ultimate Dharma" then such verses can be given, but nothing specific in them about the so-called Mlecchas. Those verses are said within the context of Hindu Dharma only.
The narrow ideas about dharma given in the post Vedic Smritis have been given in the above answer by sv. It is not necessary to follow the Smritis on this issue. (In fact all unreasonable texts whether in Sruti, Smriti, Purana, dharmasastras or Itihasas should either be properly interpreted or discarded.) There is a better definition of dharma (understood as righteousness) given in Hindu scripture.
Bhishma said, ‘..it is difficult to say what righteousness is. It is not easy to indicate it. No one in discoursing upon righteousness, can indicate it accurately. Righteousness was declared (by Brahman) for the advancement and growth of all creatures. Therefore, that is righteousness which leads to advancement and growth in righteousness. Righteousness was declared for restraining creatures from injuring one another. Therefore, that is righteousness which prevents injury to creatures. Righteousness (Dharma) is so called because it upholds all creatures. In fact all creatures are upheld by righteousness. Therefore, that is righteousness which is capable of upholding all creatures. Some say that righteousness consists in what has been upheld in the Srutis. Others do not agree to this. I would not censure them that say so. Everything, again, has not been laid down in the Srutis.’
Mahabharata, Santi Parva, Section CIX
This broader definition of Dharma is not limited only to Hindus but to all of humanity by calling all to behave in such a manner that leads to the advancement and growth of all, that prevents injury to all, that upholds all.
If by mleccha, you mean an outsider, a person belonging to a different religion or country, etc., who has no connection with the four varṇas of Hinduism, then your question maybe rephrased as:
What's the dharma of a Christian, Muslim or Jew according to Hindu scripture?
In this sense, the question is off-topic for the site. Just because mlecchas (outsiders) are mentioned in Hindu scripture doesn't mean it also discusses their duties (dharma).
Following are some excerpts from Aloka Parasher-Sen's papers on this subject and they reveal how the mlecchas were perceived by the Hindu society at different periods.
Ritual status sanctioned by the Dharmaśāstras could only be acquired by birth which foreign rulers and people did not possess. Their actual status, expressed in terms of political or economic power, stood in direct opposition to this theoretical principle. Ancient lawgivers made no explicit attempts to resolve this contradiction and nowhere do they state that acquisition of such power mitigated the fact of their being mlecchas. In other texts, however, this confusion was resolved by applying the term mleccha to foreigners in a vague and ambiguous manner which makes it difficult for us to indiscriminately label all the Yavanas, Śakas, Pahlavas, Kuṣāṇas and Hūṇas as mlecchas.
The occurrence of the word mleccha is first attested in the Satapatha Brahmana. It appears in a linguistic context and denotes unintelligible speech. However, in both Buddhist and Brahmanical literature language, or ethnic origins, or religious beliefs and rites, or the area of habitation, were never considered as mutually exclusive criteria for distinguishing, and then discriminating against, those who persisted not to accept the superiority of the dominant culture. We can, however, delineate broad areas of discrimination and distinction which the ancient Brahmanical texts emphasized with regard to the mlecchas.
In the beginning, it was a linguistic disparity that called for a basis of distinction between the aryas and the mlecchas. The early Buddhists writings distinguished between milakka and 'other' areas of habitation. In the Dharmasastras and Smritis, we have the gradual but definite evolution of the notion that aryas must make conscious efforts to avoid contact with all aspects of the mlecchas because of the latter's non-acceptance of the varnasramadharma and the non-peformance of certain brahmanic rituals.
Finally, from the early centuries A.D. the undermining of the spiritual authority of the brahmanas largely determined the discriminating statements about mlecchas. This is apparent in both Dharmasastra and non-Dharmasastra material.
[The Designation Mleccha For Foreigners In Early India, Aloka Parasher
Proceedings of the Indian History Congress, Vol. 40 (1979), pp. 109-120]
The question of mleccha behaviour began to disturb brāhmaṇa writers to a considerable degree in the early centuries of the Christian era when it is reflected in their writings. This period saw a notable increase in passages highly eulogizing the brāhmaṇas and advocating obedience to them.
The fortunes of Brāhmaṇism oscillated particularly during the period when foreign invasions actually set in and the close and convenient alliance between the kṣatriyas and brāhmaṇas was disrupted and replaced by other ruling classes. As resistance against foreign influence, the rules of exclusion for the āryas were strengthened.
The Viṣṇusmṛti has devoted a whole section to the acts which the āryas were forbidden to carry out amongst the mlecchas or in their country. They were not to offer a śrāddha in the country of the mlecchas or to visit such a country. By constantly drinking water or bathing in a pool situated in such a country, they were equated with its inhabitants. To a person defiled by speaking to mlecchas was recommended sipping water afterwards. Finally, the mlecchas were identifiable by their non-observance of caste rules, which meant that their behavioural norms in family and society were contrary to those laid down in the Śāstras. This last aspect was emphasized by Manu as well in relation to not only the mlecchas but all those who did not accept the varṇāśramadharma.
The association of the mlecchas with the disruption of the varṇa oriented society is significant and in most of the Purāṇas the Kaliyuga sees the final dominance of mleccha dharma which is in direct opposition to the "proper way of existence". As a threat to established society, it is a permanent feature during this yuga though the texts speak of it in the future tense.
Dharma is eternal and fixed.
Among the brahmanical writers Kautilya is an exception. Though he warns the King that border areas of the Kingdom, where mleccha forest tribe (term used is mlecchatavi) and bands of robbers abound, is that of enemies, he recommends at some length, how mleccha groups were to be exploited for the purposes of the State. Kautilya, like most political theorists, realized the political advantage to be gained in keeping tribes in general contented. While it is admitted that a whole army of forest tribes could pose a threat at times, the king is openly advised to secure the help of certain mlecchas for his own personal needs. Their services were to be deployed in inflicting harm upon the enemy. In another context leaders of mleccha forest troops are to be used to assassinate a weak King. As spies, men and women of mleccha communities are used to destroy enemies by poisoning them and thereby protecting the four varnas against the unrighteous. They are even trusted as enemies inside the king's palace.
[The Designation Mleccha For Foreigners In Early India, Aloka Parasher, p. 111]
Apparently, the term mlechha is also used for an offspring of a kṣatriya man and a śūdra woman. The Garuḍa Purāṇa says the following about several mixed castes.
Teachings of Yājñavalkya
1-5. I shall enumerate the mixed castes and also the duties of the householders. A brahmin father and a kṣatriya mother beget a Mūrdhābhiṣikta. Similarly,
Brahmin father + Vaiśya mother > Ambaṣṭha
Brahmin father + Śūdra mother > Niṣāda, Parvata
Kṣatriya father + Vaiśya mother > Māhiṣya
Kṣatriya father + Śūdra mother > Mlechha
Vaiśya father + Śūdra mother > Karaṇa
Kṣatriya father + Brahmin mother > Sūta
Vaiśya father + Brahmin mother > Vaidehaka
Śūdra father + Brahmin mother > Cāṇḍāla (Lowliest of all)
Vaiśya father + Kṣatriyā mother > Māgadha
Śūdra father + Kṣatriyā mother > Kṣattṛ
Śūdra father + Vaiśyā mother > Āyogava
Māhiṣya father + Karaṇī mother > Rathakāra
6-7. These mixed castes are unprivileged ones whether Anulomaja (higher caste father and lower caste mother) or Pratilomaja (higher caste mother and lower caste father). On account of the intrinsic loftiness, they will have the rights of their original caste in the seventh generation if the duty has undergone change or in the fifth generation if the duty is the same. A householder shall perform everyday the rites according to the Smṛtis in the fire first lighted on the occasion of marriage.