Partly using a combination of Family Search and Scotland's People Index - it seems certain the son Archibald McARTHUR (b 1835) at least, remained in Scotland.
HIS BIRTH:Archibald McARTHUR, 25 Mar 1835
Parents: John McARTHUR and Margaret WATT
https://familysearch.org/ark:/61903/1:1:XTGM-61M
MARRIAGE 1860:Archibald McARTHUR to Elizabeth WALKER, in Glagow on 22 Jun 1860
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https://familysearch.org/ark:/61903/1:1:XYWQ-W4Q
CENSUS 1861: At, Bathgate, West Lothian
Head: Archibald McARTHUR 26, Coal Miner, b New Monkland, Lanarhshire
Wife: Elisabeth McARTHUR 21, b Springburn, Lanarkshire
BIRTH of WIFE 1840:Elizabeth WALKER, born 30 March 1840 in Glasgow Barony
Parents: John WALKER and Margaret REID
https://familysearch.org/ark:/61903/1:1:X19L-TJL
BIRTH OF THEIR FIRST CHILD:https://familysearch.org/ark:/61903/1:1:FQS9-6XT Margaret Reid McARTHUR, b 24 Apr 1861 Bathgate,West Lothian,Scotland.
Parents: John WALKER and Margaret REID
Then they have:Mary (1863, Old Kilpatrick, Dunbarton),
Elizabeth (1865, Old Kilpatrick, Dunbarton),
John (1867, Old Kilpatrick, Dunbarton)
Robert Walker (1869 Cadder, Lanark)
Archibald (1872 Cadder, Lanark)
James (1873, Cadder, Lanark) (probably died aged 1, registered Cadder 1875)
CENSUS 1871: At Lindsay's Land, Cadder, Lanarkshire
Head: Archibald McARTHUR 36, Iron Miner, b Airdrie
Wife: Elisabeth McARTHUR 31, b Springburn, Lanarkshire
Dau: Margaret R McARTHUR 9, b Armadale, Linlithgowshire
Dau: Mary McARTHUR 7, b Old Kilpatrick, Dumbartonshire
Dau: Elizabeth McARTHUR 5, b Old Kilpatrick, Dumbartonshire
Son: John McARTHUR 3, Old Kilpatrick, Dumbartonshire
Son: Robert McARTHUR 1, b Bishopbriggs, Lanarkshire
DEATH of WIFE 1878:Of wife, Elizabeth McARTHUR, other surname WALKER, age 37
Registered in Cadder, Lanarkshire
CENSUS 1881: At 8 Eastfield Street, St Rollox, Glasgow Barony
Head: Archibald McARTHUR 46, Iron Miner, b Airdrie
Dau: Margaret R McARTHUR 19, Cotton Coop Winder, b Armadale, Linlithgowshire
Son: John McARTHUR 13, Scholar, b New Kilpatrick, Dumbartonshire
Son: Archibald McARTHUR 8, Baker, b Cadder
CENSUS 1891: At 3 Greenbank Tce, Dennistoun, Glasgow Barony, Lanarkshire
Head: Archibald McARTHUR 56, Machine Driller, b Airdrie
Dau: Lizzie McARTHUR 25, Housekeeper, b Old Kilpatrick
Son: John McARTHUR 23, iron Turner, b Garscadden, Dumbarton
Son: Archibald McARTHUR 18, Baker, b Bishopbriggs
Visitor: Maggie WATT 16, Scholar, b AirdrieCENSUS 1901: At 786 Springburn Road, Dennistoun, Glasgow Maryhill
Head: Archd McARTHUR 66, Iron Driller, b Airdrie
Dau: Mary McARTHUR 36, Baker's Shop Assistant, b Old K'ptk, Dumbartonshire
Dau: Elizabeth McARTHUR 34, Baker's Shop Assistant, b Old K'ptk, Dumbartonshire
So who is the 1891 Visitor: Maggie WATT...
perhaps you recognise her from research you already have on the WATT family?
Cheers AMBLY