Thanks for that, Ladyhawk. I'd seen the census entries and been told about the passenger lists by another researcher working the Fawcett family, but I must admit I've never seen the original passenger list documents myself, so I can't really comment on the Lizzie thing (my correspondent told me it said Susannah, but he may just have seen what he was expecting it to say)
On (the hugely improved since I last used it) FamilySearch website, I found immigration cards for an Ann Brandon [sic] and her infant daughter, Susanna, who arrived in Philadelphia from Liverpool on the ship Indiana, November 28th 1887. I suspect these might be the same people going to the US for the first time.
According to findagrave.com, Ann(a) died of consumption in Nanticoke, Luzerne County, Pennsylvania in 1893.
Susanna(h) appears in the US 1900 census, as "Susie Branton" ED26 Fell Township, Lackawanna County, Pennsylvania (sorry, I really don't know how to cite US census entries):
Matthew Branton b May 1861 39
Elisabeth A Branton b Sept 1858 41
William Dawe b May 1879 21
Jennie Branton b June 1895 5
Fred Branton b Aug 1897 2
Rena Dawe b July 1886 13
Susie Branton b July 1886 13
Beatrice Branton b. Sept 1890 9
Edward Branton b. March 1892 8
This census gives her birthplace as Pennsylvania (as does the 1891 census), but in several other documents she states she was born in England, including the documentation for her marriage on 17/6/1908 to Robert Algar. To my knowledge, she never uses the surname Fawcett.
And, yes, Ann Fawcett/Branton's maiden name was Stone. In the 1881 census, my great-great-uncle, her future second husband is lodging with her mother and stepfather in Eston (but his surname is misspelt as "Ranton")
The fact that several documents say Susie Branton was born Matthew Branton's daughter in Pennsylvania still makes me wonder if the Susannah Fawcett born at Guisborough is the same person, but presumably the certificate may give me some clues, when it arrives.
And if anyone else is researching this family, please do get in touch!