Walter Briggs was in Scituate as early as 1643, when his name appears on the list of those able to bear arms. According to Deane, In 1651, he bought a farm of Mr. Hatherly on the north side of Farm Necknad the cover there still bears the name of Brigg's Harbor."
John (Capt. John), b. Dec. 1644; m. Deborah Hawke.
James (Lieut. James), b. 1645; m. Rebecca Tilden.
Cornelius (Capt. Cornelius), b. 1647; m. (1) Mary Russell, (2) Mehitable (Allyn) Annable.
Hannah, b. about 1650; m. Samuel Winslow.
Joseph, b. about 1652.
Last Will and Testament
Dated 16th January 1676 and recorded Court 4th June 1684 lists his wife Francis, mentions his "neger" Jemy and Mariah and living at Conihassett. Also sons John (eldest) & James and land in Scituate. Daughter Hannah Winslow. Also son Cornelius and land in Swansey.
Secondary sources give Mary as the name of his first wife/mother of his children, but could not find birth record or other record of this. Briggs (1938) lists several land transactions. Possibly a church record?
"Walter Briggs was twice married. The surname of his first wife Mary, who was the mother of his children, is not known. We only know that she bore him a family of four sons and a daughter, and that she was living in 1558/9. She died not long after that date, for about 1602 Walter married, for a second wife, Frances, the widow of John Rogers of Marshfield." 
Winsor doubts that he lived in Duxbury; more likely, he says, Walter lived in Scituate.
Briggs, Samuel. A Partial Record of the Descendants of Walter Briggs of Westchester, N.y: To Which Is Added Some Account of His Ancestry, Collateral Branches, Origin of the Family Name, Ancient Pedigrees, Wills, Etc, Etc. Cleveland, Ohio: Printed for private circulation only by Fairbanks, Briggs & Co, 1878.
Sherman, Robert M, Verle D. Vincent, Robert S. Wakefield, and Lydia D. Finlay. Mayflower Families Through Five Generations: Descendants of the Pilgrims Who Landed at Plymouth, Mass. December 1620. Plymouth, MA: General Society of Mayflower Descendants, 1997. Vol. 15: Chilton/More, p. 16.