Hello Fran, No relation or connection. Had a look to see if I could find your Minks. I believe I found Elizabeth's family in 1871 census. 1871 census for Tranmere, Cheshire Minks family living at No 91 Greenway Rd.
John MINKS. Head. 55. Cow Keeper. born Bebington,Cheshire. Sarah. wife 46.b.Hawarden, Flintshire, Wales. Peter. son 20. b.Bebington. Charles 17. b.Bebington. George.14 b.Bebington. Thomas. 11. b.B'ton ELIZABETH 6.b.Bebington Elizabeth Bennett. Niece.15.b.Hawarden, Flintshire. ------------------------- Sarah 56 was a Widow by 1881 census and living at No.18 Prenton Lane, Tranmere, Cheshire. Says she was born in Buckley, Flintshire. Elizabeth was also with her in 1881 aged 16 +other sibs.
Elizabeth married Charles Thomas Brine in 1883. Marriage reg in Birkenhead, Cheshire.
I think John Minks married Sara WINN in 1849. Marriage reg in Liverpool. The name was missspelt in 1851 census. I found John and Sarah under Mynl on the indexes to census. The name was again transcribed incorrectly as on the census it was written MYNK but could be taken for Myln. John 35. Shoe Maker b.Cheshire. Sarah 28. b.Wales. Peter. son a few months old. b. Tranmere,Cheshire There was also a William Mynk aged 12. Shoemaker's App. It's poss that John was previously married and that Sarah was his 2nd wife. I haven't looked any further.