Doesn't appear to have been an historically strong route. Will add more detail as I find it:
Northern Air LinesBelfast ( Malone ) - Carlisle
Mail & newspapers from 2 September 1924 until the winter, once daily with DH.50 and DH.9
Routed across the Solway Firth, along the coast to Stranraer and then straight across to coast-in at Larne.
From 3 March 1925 it seems to have resumed with an intermediate stop at Freugh to collect mail from the Stranraer train.
Edit: some more information
here which states that the first service period only lasted a month.
BEA Belfast ( Nutts Corner ) - Carlisle - Newcastle
1 July until early October 1947 with a DH Rapide, twice daily except Sunday, for monthly total of 520 seats.
Prices:
| Newcastle - Carlisle | £1 5s |
| Newcastle - Belfast | £3 19s |
| Carlisle - Belfast | £3 |
And a runner-up:
West Coast Air Services
Newtownards - Isle of Man
From April 1935 with a DH Dragon, they also flew Carlisle - Isle of Man later in the day but don't seem to have linked the two services.
Another Belfast - Carlisle aviation link was between Shorts and CSE Aviation. CSE were the UK distributor for Piper PA-38 Tomahawk and initially assembled the aircraft from kits at Carlisle, prior to the planned production of the type in Belfast. But that idea was abandoned in 1980 because the FAA declined to extend their certification over a Shorts-built version being sold outside the USA, so they would have needed to recertify under the CAA and other authorities.