Some records exist for the Ballinderry Moravian church. There’s a copy in PRONI in Belfast. A personal visit is required to view them. This is what they contain:
Baptisms, 1754-1817; marriages, 1784-1800; register of members, 1755-1816; ministers’ diaries, 1768, 1803-04, 1810-17 and 1842-90, which include baptisms, marriages and deaths, and lists of members; history of the church from 1750 and published 2000.
Here’s a family in the 1851 census with James & Elizabeth as head of household, so that could well be your family:
http://www.census.nationalarchives.ie/pages/1851/Antrim/Upper_Massereene/Ballinderry/Ballinderry/12/
Here’s James & Sophia’s marriage in Ballinderry Church of Ireland in 1878 (His surname was written as Fleeting):
https://civilrecords.irishgenealogy.ie/churchrecords/images/marriage_re…
This looks to be your family in 1901, by which time they were living in Belfast:
http://www.census.nationalarchives.ie/pages/1901/Antrim/Shankhill/Upper_Meenan_Street/950592/
And in 1911:
http://www.census.nationalarchives.ie/pages/1911/Antrim/Shankill/Agnes_Street/150224/
In 1911 they had 11 children of whom 9 were still alive. You could search the 1911 census for the ones that are missing, and also the irishgenealogy site for the 2 deaths and possible marriages for the others. The marriage and death records cover up to 1921, after which you need to switch to the GRONI site. Only they have records for 1922 onwards.
I don’t see any mention of a Moravian church in Belfast so not sure where they attended when living there.
Here’s James’s death on 8th Jan 1915:
https://civilrecords.irishgenealogy.ie/churchrecords/images/deaths_returns/deaths_1915/05283/4467797.pdf
Sophia’s should be on the GRONI site (£2.50 to view).
I noted a few probate abstracts for Fleetons whose forenames match those in the 1901 census. I do not know if they are your family. You would need to research further, by looking up their death certificates, wills etc, but they may give you a clue as to what happened to some of the children. Fleeton is not a common name in Ireland and even if they are not your immediate family, they may be related:
Fleeton, Samuel Alexander of 9 Skyline Drive Harmony Heights Lambeg county Antrim post office electrical engineer died 20 November 1964 Administration Belfast 5 January to Mary Anna Fleeton the widow. Effects £2300.
Fleeton Elizabeth of Tullyballydonnell county Antrim spinster died 24 January 1941 Probate Belfast 27 February to David Thompson grocer. Effects £397 2s. 2d.
Fleeton, William John of 84 Upper Charleville Street Belfast plater died 8 May 1968 at Musgrave Park Hospital Belfast Administration Belfast 10 June to Charlotte Fleeton widow. Effects £816 19s.
This might be Elizabeth Fleeton's death in Belfast in 1887. Informant was son James:
https://civilrecords.irishgenealogy.ie/churchrecords/images/deaths_returns/deaths_1887/06227/4782062.pdf
If it is the right lady, then she was a widow so William didn’t die din1895. However I don’t see any death in 1895 and I do see one on 22.12.1874 in Lisburn (aged 64) so I suspect that’s when he died. That year isn’t on-line free yet but you can view the death cert on the GRONI site.