I don't know for Kaliyuga, but we can look at when the Manusmṛti applied, because it is likely to be somewhat close to the truth of the matter from then to now.
First of all, everyone (possibly means every Āryan or every human to make some later verses make sense) back then was equal to a Śūdra if they are not "born in the Veda."
नाभिव्याहारयेद् ब्रह्म स्वधानिनयनाद् ऋते ।
शूद्रेण हि समस्तावद् यावद् वेदे न जायते ॥ १७२ ॥
nābhivyāhārayed brahma svadhāninayanād ṛte |
śūdreṇa hi samastāvad yāvad vede na jāyate || 172 ||
He should hot pronounce Vedic texts, apart from the Svadhā-offering; because so long as he is not born in the Veda, he is equal to a śūdra.—(172)
These verses are for constant interbreeding, but they show an important point with the word "within," which shows the process of passing on Varṇa has a lot of randomness in it, so the Manusmṛti cannot say how many generations, just an upper limit.
शूद्रायां ब्राह्मणाज् जातः श्रेयसा चेत् प्रजायते ।
अश्रेयान् श्रेयसीं जातिं गच्छत्या सप्तमाद् युगात् ॥ ६४ ॥
śūdrāyāṃ brāhmaṇāj jātaḥ śreyasā cet prajāyate |
aśreyān śreyasīṃ jātiṃ gacchatyā saptamād yugāt || 64 ||
If the child born from a Śūdra woman to a Brāhmaṇa goes on being wedded to a superior person,—the inferior attains the superior caste, within the seventh generation.—(64)
शूद्रो ब्राह्मणतामेति ब्राह्मणश्चैति शूद्रताम् ।
क्षत्रियाज् जातमेवं तु विद्याद् वैश्यात् तथैव च ॥ ६५ ॥
śūdro brāhmaṇatāmeti brāhmaṇaścaiti śūdratām |
kṣatriyāj jātamevaṃ tu vidyād vaiśyāt tathaiva ca || 65 ||
The Śūdra attains the position of the Brāhmaṇa and the Brāhmaṇa sinks to the position of the Śūdra; the same should be understood to be the case with the offspring of the Kṣatriya or of the vaiśya.—(65)
Anyway, back to the question, this verse suggests someone with a changed Varṇa passes it on:
योऽनधीत्य द्विजो वेदमन्यत्र कुरुते श्रमम् ।
स जीवन्नेव शूद्रत्वमाशु गच्छति सान्वयः ॥ १६८ ॥
yo'nadhītya dvijo vedamanyatra kurute śramam |
sa jīvanneva śūdratvamāśu gacchati sānvayaḥ || 168 ||
The twice-born man, who, not having learnt the veda, labours over other things, soon falls, along with his descendants, even while living, to the state of the śūdra.—(168)
Most importantly, this verse suggests all Brahmaṇa change Varṇa, because they are not born from out of the womb as Brahmaṇa. Thus changed Varṇa has to matter or all of the verses related to having a Brahmaṇa father would make no sense:
तत्र यद् ब्रह्मजन्मास्य मौञ्जीबन्धनचिह्नितम् ।
तत्रास्य माता सावित्री पिता त्वाचार्य उच्यते ॥ १७० ॥
tatra yad brahmajanmāsya mauñjībandhanacihnitam |
tatrāsya mātā sāvitrī pitā tvācārya ucyate || 170 ||
Among these, at that Brahmic birth which is. marked by the tying of the girdle, the Sāvitrī has been declared to be his “Mother,” and the Teacher his “Father.”—(170)
Karṇa actually has one of the highest births possible (if what is said by the Manusmṛti still applied then), so people need to stop thinking that he represents a marginalized group.
क्षत्रियाद् विप्रकन्यायां सूतो भवति जातितः ।
वैश्यान् मागधवैदेहौ राजविप्राङ्गनासुतौ ॥ ११ ॥
kṣatriyād viprakanyāyāṃ sūto bhavati jātitaḥ |
vaiśyān māgadhavaidehau rājaviprāṅganāsutau || 11 ||
One born from the Kṣatriya on the Brāhmaṇa maiden is ‘Sūta’ in caste; and the sons born on the Kṣatriya and the Brāhmaṇa maiden from the vaiśya are ‘Māgadha’ and ‘Vaideha’ respectively.—(10)
See comments for a more in-depth explanation if you still doubt this. In short, a Sūta is definitely between a Kṣatriya and a Brāhmaṇa (closer to a Kṣatriya) in order to make the second and third verses I quoted make sense and their portrayal in scriptures such as the Mahābhārata confirm this.