Here are some news article examples containing 'with my/their etc. being...' as supplement to a main clause:
(1) Since the opposing counsel would be the U.S. Department of Justice, and with my being new to that game, I enlisted as co-counsel an accomplished federal litigator with extensive experience in dealing with the DOJ. (Source)
(2) Since this raised many leftists' ire and with my being the reasonable man I am, I propose a compromise: no immigrants from the Third World or the Old World. In other words, no immigration, period. (Source)
(3) The Jordans were serving at Destiny Life Church in Oakland, and with their being hired by Divine Hope Church, both churches merged while keeping the Divine Hope name. (Source)
Is the use of genitive pronouns (my, their, etc.) well-formed and natural?
Or is it better to have accusative pronouns (me, them, etc.) instead?
In a related question "with/without + pronoun (me vs. my) + gerund-participial phrase", I have shown this example from The Cambridge Grammar of the English Language (Page 461):
We set off again, the Rover going precariously slowly in very low gear up hills, with me staying on its tail in case it petered out altogether.
CGEL says, and I agree, that my cannot replace me in this sentence.
If the above examples (1)-(3) are well-formed and natural, how do you distinguish (1)-(3) from the CGEL example?