Frank re-married Mary ?? (born c 1859, Yorkshire) before Apr-Jun 1873 and they had five children, three of which were born in Hull - Arthur Apr-Jun 1873, Nora b. Jan-Mar 1875 & Kate Jan-Mar 1877.
Nora & Arthur were both in the Leicester Chronicle on 13 June 1891.
Arthur was described as a shoe hand of no fixed residence and was accused of breaking into the Leicester Co-operative Society at Upper Conduit Street on 5th June and stealing 4 & 1/2d plus a pork pie worth 8d. He was sentenced to a months hard labour, but the article also refers to his father having to go to Lincoln, where his daughter was charged with a felony.
Further down the same page/column there is then a report about Norah Mitchell aged 16 daughter of Frank Mitchell, assistant at the Leicestershire Dispensary, charged with stealing a fur cape worth £1 at Lincoln. The article also makes reference to her Grandfather at Hull, where she had been staying for the previous 4-months.
I haven't looked for any of the other children, but based on the above you seem to be correct about Frank's movements.
