One portion of the Ainsworth family sailed to New England and settled there. This branch of the family was fairly well documented (to that point) in the 1894 book “Genealogy of The Ainsworth Families in America”, by Francis J. Parker. Microfilm copies of this book can be obtained through any LDS Family History Center.
Unfortunately, the Parker book neglected most of the Southern Ainsworths except one (James Ainsworth b 1785). The Southern Ainsworths appear to be descended from someone who sailed to Virginia, which received far more Lancastrians than New England.
New England Ainsworths are, nevertheless, welcome to join the Ainsworth discussion group and network your information with ours.