On 1870 census for La Salle, Illinois, there is a John Plimmer, age 40, with Wife Mary, 40 both born England.
Sons:
Richard (9 or 4... sorry hard to decipher) born England.
Benjamin, age 5 born PA
Samuel, age 2, born Illinois
If this is your family, it may help narrow down the arrival date and where. Unable to assist with passenger lists.
Valerie