I agree with others that a GFCi wouldn't trip on a loose connection.
A loose connection can produce enough heat to start a fire without
arcing (a "glowing" connection). Late stages in failure are likely to arc.
AFCIs sense "parallel" arcs - from hot-to-neutral (a "fault", the F in
AFCI). Starting in 2008 the NEC requires them to also detect "series"
arcs, as in a loose connection. As far as I know, no current AFCIs
detect series arcs, so none would detect a loose connection.
Methos
A GFIC is made to trip on sudden line voltage changes. A slow warm up
would not trip the built in breaker. This very problem cause a house
fire that burned a third of my parents house to the ground. Two wires
in some old Romex touched and caused a slow short but it was not enough
to trip the breakers. Therefore, it warmed up long enough to catch fire.
GFIC's are a nice safety feature but it certainly doesn't catch all
problems
GFCIs trip on a difference in current between the hot and neutral, not
line voltage changes. They are primarily for electrocution protection.
The "slow short" you describe, an arc that trips a breaker slowly, if at
all, is exacty what AFCIs are designed to protect against. A more likely
cause is probably an abused extension cord.
AFCIs also include 30mA ground fault protection (GFCIs have 5mA
protection). The idea is, I think, that if a ground wire is adjacent, a
hot-to-neutral arc is likely to also become hot-to-ground.
bud--