I am not a Beckwith family member and wasn't a Beckwith researcher until about an hour ago
I don't think you are going to get any help on the American side. The most reliable source is probably The Ancestry of Emily Jane Angell, by Dean Crawford Smith and Melinde Lutz Sanborn, (New England Historic and Genealogical Society, 1992). The information on Matthew Beckwith begins on page 223. (The NEHGS is the gold standard for New England genealogy research.)
They say Matthew was born in England-that's it. They acknowledge a rumored connection to Yorkshire but say there is no proof.
you can download a .pdf of the book here:
https://archive.org/details/ancestryofemilyj00smit#donate_dropdownVery nice of them to make it publicly available, by the way, despite being under U.S. Copyright.
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This researcher, whom I would trust next since he is also connected to a genealogy society, says Matthew was born in Yorkshire but he gives no source.
"Matthew Beckwith, Hartford Founder" (no date)
https://www.foundersofhartford.org/the-founders/matthew-beckwith/---
The information at findagrave, which mentions Pontefract, seems to be based on the images attached to the memorial, which come from familysearch.
https://www.findagrave.com/memorial/34387850/matthew-beckwithBut see the warning which someone added to the familysearch tree that says most of the information is incorrect:
https://www.familysearch.org/tree/person/details/LB86-MDM-----------
It's too bad, since if you knew where Matthew was from, then you would be pretty sure you knew where your 4x great grandmother was from. I'd try to work back from Elizabeth and see if you can make any progress that way.
I just noticed that you said the DNA shows "very clearly" that you are connected with Matthew Beckwith from New London, Connecticut. I don't claim to know anything about DNA matching, but could you have a 'very clear' match if the common ancestor was 400 years ago? Could it be that one of Matthew's descendants travelled to Madras and the connection is later?