They really were rotated through the fleet as described. The only difference now is that the flying fleet of T1s is very small compared to the days of Chiv + Brawdy + Valley + Scampton + assorted other users, so they typically come from storage these days. The modifications for RAFAT use are relatively minor and are detailed in the aircrew manual so any pilot can ferry them if required.
When I was doing my training at Chiv the fleet was painted in air defence grey, but every so often an ex-red Hawk would be rotated and absorbed into the TWU fleet, giving you quite a disadvantage in a visual fight if you were allocated the bright red jet.