Parish Records St. Dunstan and All Saints, Tower Hamlets, London
Christening November 19 1758 Francis, 35 days old, son of Francis Walton, Ratc[liffe], Mariner and Amy

London England tax Records 1799
Wapping Street, Proprietor Samuel Clark, Occupier Francis Walton.

From a family member.
Francis Walton, was a sea Captain, and one of his more noted voyages was as captain of the "Friendship", one of the First Fleet taking convicts to Australia.

First Fleet Fellowship Victoria Inc - Friendship Convict Transport and Captain Francis Walton
https://firstfleetfellowship.org.au/ships/hms-friendship/

A topographical dictionary of London and its environs: by James Emes 1831
London Docks, They are situated at Wapping, and arose from the want of sufficient harbour room in the Port of London. The eastern, or St. George’s dock, occupies more than 20 acres of what was formerly Wapping, and measures 1260 feet in length, 890 feet in width, and 27 feet in depth. this dock covers the space from Virginia-street almost to Old Gravel-lane, and is capable of holding 300 ships. The new dock is eastward of the preceding, and will hold about 50 ships. St. George’s dock was opened on the 1st of February 1805, when the Perseverance of Liverpool, the oldest vessel in the Oporto trade, decorated with the flags of all nations, sailed into the dock which is surrounded by a quay 100 feet wide, and by large stacks of warehouses.
This commercial company ... Captain Denis Chapman, Superintendent; and Captain Francis Walton, Dock-master.

Another source says he was appointed dock master in 1804.

In the Holdens Triennial Directory 1805
Captain Francis Walton, 284 Wapping

In the London City Directory for 1811
Walton Francis, dock master of the London Docks, 284 Wapping St.

In the January to March 1840 death index Francis Walton, Stepney, London, volume 2, page 359

Parish records St. John of Wapping, Tower Hamlets, Middlesex
January 4, 1840 burial Francis Walton of Pier Head, Wapping in his 82nd year.

Parish Records St. Ann Blackfriars, London
Baptism June 5 1771 Kitty daughter of William and Rebecca Cock

From a family member.

London Marriage Bonds Allegation
June 27, 1793
Appeared personally Francis Walton and made Oath, that he is of the Parish of Saint John Wapping in the County of Middlesex a bachelor of the Age of Twenty one years and upwards and intended to marry with Kitty Cock of the parish of Saint Leonard Shoreditch in the County of Middlesex a spinster of the age of twenty one years and upwards and that he knoweth no lawful Impediment, by Reason of any precontract, Consanguinity, Affinity, or any other lawful means whatever, to hinder the said intended Marriage and prayed a licence to solemnize the same in the Parish Church of Saint Leonard Shoreditch aforesaid an further made Oath that the usual Place of Abode of her the said Kitty Cock hath been in the said Parish of Saint Leonard Shoreditch for the Space of four Weeks last past.
Signed Francis Walton
Sworn before me S. P. Parson
Surrogate

Parish Register St Leonard, Shoreditch, London, Hackney Borough
Francis Walton of the Parish of St. John Wapping Middlesex Bachelor and Kitty Cock of this parish Spinster were married in this church by Licence on the 29th day of June in the year One Thousand Seven Hundred and Ninety Three by me Joseph Rose? Minister in the Presence of Samuel Clarke and Amy Walton

In the 1841 census Wapping Pierhead No. 5, St John Wapping parish, Ossulstone hundred (Tower Division), Middlesex, Stepney registration district, Shadwell and Wapping sub district.
Kitty Walton, age 70, Independent, born in county.
Amy Walton, age 25, born ib county.
William Walton, age 35, solicitor, born in county.
4 others, 2 of which are female servants.

In the 1851 census Kitty Walton, head, widow, age 79, Fund Holder, born London.
Daughter Amy Walton, unmarried, age 34, Fund Holder, born Wapping, Middlesex.
Son William Walton, unmarried, age 46, Solicitor, born Wapping.
A former servant age 93, an attendant companion and 2 other servants.
[Next door was the dockmaster, a successor to Francis Walton]

In the April to June 1852 death index Kitty Walton, Stepney, London, volume 1c, page 285.

Parish register St John of Wapping, Tower Hamlets borough, Middlesex burial
Kitty Walton of Pier Head Wapping, April 7 1852, in her 81st year.