With due respect, On what account were the Kuru lineage after Dhritrashtra and Pandu considered Kshatriya varna in Mahabharata when Dhritrashtra and Pandu themselves were born from a brahmin sage vedvyas ?
From the the link of this section of Adi Parva of Mahabharata which OP provided in comments:
In a former millennium, Parashurama, killed Kartavirya Arjuna in battle for the sin of slaying his father. Not only did he kill the followers of Kartavirya Arjuna, but he destroyed the entire kshatriya race twenty-one times. When the earth was devoid of great warriors, the queens approached the great rishis and procreated children by them and thus the kshatriya race was revived. Therefore, a purified brahmana should be invited with an offer of wealth, and let him raise children by the wives of Vicitravirya.
So, it's clear that kshatriya race was procreated by rishis/brahmins long back also.
This was due to a practice known as Niyoga.
There are 12 kinds of sons and son born using the Niyoga practice known as kṣetraja son:
यस्तल्पजः प्रमीतस्य क्लीबस्य व्याधितस्य वा ।
स्वधर्मेण नियुक्तायां स पुत्रः क्षेत्रजः स्मृतः ॥ १६७ ॥
If a son is born of the wife of a man, either dead or impotent or diseased, by one who has been duly ‘authorised,’—that son is declared to be ‘kṣetraja’ ‘soil-born.’ ~Manu Smriti 9.167
From above references, it's clear that even though Dhritrashtra and Pandu were biological sons of Vyasa but still they would be considered kṣetraja sons of Vicitravirya due to Niyoga practice and would be considered kshatriyas ultimately.
One's varna is not decided solely based on birth. Please see following Manusmriti verses which says Education is true birth
2.147: The birth that happens from womb of mother after parents desire for procreation is an ordinary birth. Real birth happens when the person completes his education
2.157: A Brahmin devoid of education is equivalent to an elephant made of wood or a deer made of leather. They are merely namesake and not real.
10.4: Brahmin, Kashtriya and Vaishya take second birth after education. Shudra who could not complete education is fourth Varna. There is no fifth Varna among Arya or noble people.
Problems started when our ancestors replaced meritocracy with foolish birth based caste-system.
Our scriptures are laden with examples of varna migration. Let's take the example provided here
Now sons of Satyavati (Shudra) and Shantanu (Kshatriya), Vichitvirya and Chitragandha , would be classified as Ugras as per Manusmriti
10.9 From the Kṣatriya on a Śūdra maiden is born a being called ‘Ugra,’ of the stuff of the Kṣatriya and Śūdra, cruel in his deeds and dealings.—(9)
Furthermore the prescribed activities of Ugras are
10.49 For Kṣattṛs, Ugras and Pukkasas, the killing and catching of animals living underground; for Dhigvaṇas, work in leather; and for Veṇas the beating of drums.—(49)
We know for sure Vichitravirya and Chitrāngada were not involved in such activities. They were kshatriyas after they completed their education.
The same is extended to Vichitravirya's sons - Dhritarashtra and Pandu. They too became kshatriyas post education. Yes, they were conceived through Niyoga by VedaVyasa and thus are considered sons of Vichitravariya but they got assigned kshatriya varna after education