Nearly all cases like this are caused by problems with the GFI itself - usually moisture penetration. It only takes a thin coating of moisture on the plastic at the plug end (between hot or neutral blade and ground pin) to create enough current differential to trip the GFI. Drying the plug and the face of the outlet and having a good in-use cover could solve the problem from this cause.
Such moisture problems often arise with very wet weather or high humidity combined with cool nights.
The second most likely cause is a faulty GFI - they sometimes need to be replaced.
A transformer problem is possible but unlikely - there would need to be current flowing from neutral or hot to ground. Only possible if a wire has come loose and is contacting the cabinet or an energized conductor. (It is a very old transformer!) If this is the problem - be very careful - a 120-v energized cabinet is bad news.
A short circuit in a fixture is highly unlikely to trip a GFCI since both the hot and neutral secondaries are isolated from hot and neutral on the primary. The secondary neutral, on the other hand, is connected to the ground and shared with the primary ground. So, in a scenario where a short has occurred in a fixture, there may be current flowing to ground, but I don't think the GFCI would trip - it should still see a balance between the primary neutral and hot - since those are not connected to the secondary. (I could be wrong about this.)