The 1841 census of Kent, England shows
John Lambkin, 1815, parish of Boughton Monchelsea
Sarah, 1818
John, 1840
The 1851 census of England shows
John Lamkin, head, 35
Sarah, wife, 31
John, 11
William, 10
Ann, 8
Mildred, 6
Alfred, 4
Daniel, 3
George, 1
1858, Port of New York, ship American Eagle, passenger list 8 January 1858 from London:
Jane Lambkin 45 F US, Anne Lambkin 14 F England, Mildred Lambkin 13 F England, Ellen Lambkin 11 England, Alfred Lambkin 10 England. (We lost John, William, Daniel, George, and we gained Ellen. William reappears later. We later see a Nancy, too, about the same age as Ann.)
July 1860 census, John Lamkin (45), Laborer, + Jane (48) with children Alfred (14) and Ellen (16), all from England, all living in Pomfret, Fredonia, Chautauqua, New York.
The 1870 census of Portland, Chautauqua, NY shows John Lamkin (born about 1816 in England), farmer, living with Sophia Bartholomew (age 47) (housekeeping) (also from England) with children William F (9), Minnie (4), and Nellie (2), all surnamed Bartholomew.
The 1880 census of Fredonia, Chautauqua, NY shows John Lamkin, age 63, living with Sophia, age 45, with stepdaughter Minnie Bartholomew, age 12, stepdaughter Nellie Bartholomew, age 10, and son Henry Lamkin, age 6. (My guess is Sophia was 37, not 47, in 1870.)