The entry that Falkryn posted for 1861 shows the following places of birth:
Mother Elizabeth: Larbert, Sterlingshire
Robert - Old Monkland, Lanarkshire
James - Old Monkland, Lanarkshire
Elizabeth - Kilwinning, Ayrshire
I've just spotted another family in 1861

which is closer to the information you gave at the beginning:
Elizabeth Campbell 48, spirit dealer, b. Kilmarnock
John Campbell 23, cabinet maker, b. Kilmarnock
James Campbell 16, Ap House Printer, b. Kilmarnock
Address: 98 Portland St, Kilmarnock
There are a mixture of records showing on IGI (submitted and actual extract) so hard to verify children. In the main the children of Robert Campbell and Elizabeth Hart were those showing for the family in the 1851/61 entries we posted earlier.
The submitted entry on IGI for a James born in 1845 in Kilmarnock, who married a Sarah Russell in 1872, shows mother as Elizabeth Hart but wonder if the place of birth has been confused (given it is a submitted entry) with the other James I have now posted in 1861?
You should be able to verify the line through other documents (marriage certs and death certs.) which will include father's occupation and mother's maiden name. Also census data after the children that you are following married will also let you confirm places of birth.
Monica