Q: For Gaudiya Bhakts what is the Parampara? Very many descendants of Mahaprabhu are there and each professes to be the correct link to Him.
Your use of the preposition “the” in the phrase “each professes to be the correct link” isn’t exactly correct, because “the” means “the only one” or “the most important one.” There is more than one active, valid, paramparā lineage, so the correct phrase should be “each professes to be a correct link to him.”
Some lineages may indeed profess to be “the” link to Mahāprabhu (i.e. the one and only), but in so doing they prove they are not even “a” link.
Q: I had read that we are in Braham Madhav Gaudiya Sampradaya. Is this Madhvacharya or Madhvendar Puri who is Chaitanya Mahaprabhu’s Guru’s Guru.
We need only trace our lineage to Mahaprabhu, who is Svayam Bhagavān Śrī Krishna’s Audarya Rūpa. But for the sake of others who may not accept this, we can note that Śrī Caitanya accepted dīkṣā in the Madvācārya paramparā (from Śrī Īśvara Puri disciple of Śrī Mādhavendra Puri), which is a part of the Brahmā paramparā which is one of the four Vaiṣṇava Paramparā (Brahmā, Śrī, Catusana, Śiva).
Philosophically speaking, the Gauḍīya Paramparā accepts key elements from all four Vaiṣṇava Paramparā.
As for the development of the Gaudīya line after Śrī Caitanya, you can read a lot about that in Caitanya Caritāmṛta (Start at Adi-līlā Chapter 10). Gadādhara Paṇḍit, Nityānanda Prabhu, Advaita Prabhu, etc. all have important branches in the lineage from Śrī Caitanya. All these branches have their own branches, and so on.
Q: But then I think from Jagannath Das babaji there are many divergent branches.
The problem is not from Jagannatha Das bābājī, exactly, but from Śrī Bhaktisiddhānta Sarasvati. His followers accept that he took dīkṣā from Gaurakiśor dās Bābājī, but other Vaiṣṇava’s do not agree.
There is also a problem with his father, Śrī Bhaktivinode Ṭhākur. He definitely took dīkṣā from Śrī Bipin Bihari Gosvāmī, but Bipin Bihari Gosvāmī, after the passing away of Śrī Bhaktivinode, disowned him as a disciple, apparently due to disputes about the location of Śrī Caitanya’s birthplace.
Because of this, and because of his well-documented strong dislike for the Gosvāmī from the start, Bhaktisiddhānta refused to acknowledge the Goswāmī as his father’s guru, preferring to cite Śrī Jagannātha dās Bābājī instead.
This is a small problem in comparison to the whole Gauḍīya paramparā tree, but it potentially affects many people because the world was introduced to Gauḍīya bhakti mainly through disciples of Śrī Bhaktisiddhānta, especially Śrīla Prabhupāda, who founded ISKCON.
Q: Is being somewhere in this lineage also important or just learning and realising and experiencing the Knowledge would be sufficient?
Learning, realizing, and experiencing is what is important.
But it is not likely that you’d be able to do that very well without learning it somewhere in the lineage that passes on the realization and experience.
Lineage is not a technicality, because the knowledge, realization and experience is passed down by lineage. Thus, if you study Gauḍīya Siddhānta in detail you will see that the unclear dīkṣā lineage of Gauḍīya Maṭha and ISKCON is not just a technicality. It has an obvious ramification on the clarity of their realization and experience: They are almost like a separate world from the rest of the lineage of Gauḍīya bhakti, with many philosophies and practices that are entirely their own and not shared at all with the rest of the lineage.
Q: One set of people may not recognise the other set of people’s lineage, but supposing Sri Chaitanya Maha Prabhu accepts all these linkages to Him…
Lack of recognizing plural paths is not a symptom of true Vedic culture. Quite the opposite.
As expected, then, lack of recognizing other branches is not common in the Gauḍīya paramparā. It is, however, a pervasive phenomenon in ISKCON and Gauḍīya Maṭha. Even amongst themselves the “we are the only bona-fide group” syndrome is rampant. Often one Gauḍīya Maṭha branch will not visit the other Gauḍīya Maṭha branch. ISKCON will not cooperate with or recognize Gauḍīya Maṭha leaders. Etc. Etc. Even the micro-branches within each micro-branch in these groups devote a lot of time and energy to defaming the other nano-branches.
When anyone points any of this out, like this article does, the ISKCON / Gauḍīya Maṭha claims it is an “attack.” It is really a sad, psychologically complex situation.
Q: …is it Ok to switch from one lineage to the other?
No, not really.
But if you are having problems in one branch for some reason, they will help you; and if they can’t, they will help you find a better situation in another branch. If they don’t, you know they are not a genuine spiritual lineage in the first place, and then there is no problem “switching.”