I travelled to DXB on LH in May.
There was a special immigration lane for F and C pax on LH, KLM and other carriers BUT it was by invitation only, i.e. you need fast lane pass from the flight crew or ground services upon arrival (the latter may be very difficult). I saw the majority of (angry) C class pax being turned away because they had no pass/invitation (obviously, they were not aware that they should have asked for it on the plane).
The imm. lines are very long in the late evening with many flights arriving around the same time, including LH and KLM.