The family are still in Lincoln in 1871.
Reference RG10, Piece 3371, Folio 64, Page 5
Witham Bank, Lincoln
Walker, Joseph Head M 34 Labourer b Lincoln
Walker, Eliza Wife F 30 b Lincoln
Walker, William Son M 10 b Sheffield, Yorkshire
Walker, Jane E Daughter F 7 b Lincoln
Walker, Emma Daughter F 5 b Lincoln
Walker, Harry Son M 1month b Lincoln
In 1881, William is a Navvy, on the railways?
Reference RG11, Piece 3241, Folio 21, Page 35
9 George Street, Lincoln
Green, George Head M 26 Excavator on Railway, Navvy b Selby, Yorks.
+ wife, 3 children & neice
Walker, William Lodger M 20 Excavator on Railway, Navvy b Sheffield, Yorkshire
The rest of the family are in Manchester:
Reference RG11, 3908, 115, 21
6 Ernest Street, Gorton
Walker, Joseph Head M 46 Carter b Lincoln
Walker, Eliza Wife F 40 b Lincoln
Walker, Emma Daughter F 15 Scholar b Lincoln
Walker, Harry Son M 11 Scholar b Lincoln
Walker, Arthur Son M 9 b Lincoln
Walker, Joseph Son Scholar b West Gorton, Lancashire
Walker, James Son M 1 b West Gorton
So, the family moved to Lancashire after the birth of Arthur in 1872, and before the birth of Joseph in 1875?
Jane is also in Gorton:
Reference RG11, 3905, 70, 13
Crossley, William Head 54 Pawnbroker b Todmorden, Yorks.
+ wife & daughter
Walker, Jane Elizabeth Servant F 17 Domestic Servant b Lincoln