Hello Rita, The connection to the Irish and Rotherham , Yorks, Swift's can be found in "Burkes Irish Family Records" and goes thus:-
Rev Thomas Swift of Goodrich and Bridstow in Herefordshire.(1595-1658) married Elizabeth Dryden and their six recorded sons are Godwin,Dryden, Thomas,William,Jonathan and Adam.
The third son Thomasmarried Maria Davenant and they had a son (Rev) Thomas Swift who became Rector of Pattenham in Surrey.(1665-1752)
The next part can be found in IGI. Rev Thomas Swift married Ann(?) and they had several children among whom were:- 1)William Swift christened 22 Jan 1697 (Rev Swift of Hanover. VA) married Dianah Hodgkin. 2)Davenant Swift. christened 3 Mar 1701 3)Godwin Swift christened 10 Apr. 1705 5)Dean Swift christened 22 July 1706 6)Ann Swift. christened 19 May 1709. (All born in Pattenham/ Puttenham, Surrey)
Compare those names with the earlier family and you can have no doubt that they are all of the same line. It is all there in IGI, you just have to know what you are looking for !
Note: the Jonathan Swift shown above was a Dublin Lawyer and the father of Jonathan Swift the author (Gulliver's etc)and later Dean of St Patricks. He died whilst the young Jonathan was still in the womb.
Elizabeth Dryden (above) was closely related to Dryden the poet and Dryden and Jonathan Swift were cousins.
Jonathan Swift and Rev Thomas Swift of Pattenham started at the university of Dublin the same day (24th April, 1682) and Jonathan always referred to him as "My country parson cousin"
Sir William Temple was Jonathan Swift's mentor and his estate was at Moor Park near Farnham in Surrey where Jonathan spent several years. It is just spitting distance from Pattenham and Jonathan and Rev Thomas must have spent a good deal of time together.