Asa's father Stephen Dudley was born 2 Ju. 1735 Littleton, Middlesex, MA He died in South Carolina 1784. Stephen served in the Rev. War and married 21 Dec. 1758 in Littleton, MA Miss Lydia Harwood b. Feb. 1739. She d. 6 Aug. 1810 Bennington, VT. (d/o Joseph Harwood and Lydia Brooks).
Stephen and Lydia had 9 children all born in Littleton, MA with Asa being the youngest born in June 1782.
Not much else is found on that Asa so I suppose he never married or at least died without issue? Asa's older brother Gen. Peter 1773-1847 was in the War of 1812 and lived in Oppenheim, NY. Asa's oldest brother Stephen 1760-1826, lived in Cato, Cayuga Co., NY. married and had children. That Stephen's son Asa b. 1786 was my gr. gr, gr grandfather.
Good luck with your search.