Find an electric repairer / shop. Sort of place that will bench test and get them to check it out.
If your up to the job- take the fitting down.
Test for continuity from the transformer to each of the 12 bulb fittings.
Wire the transformer to mains, test the output side for the transformed voltage- 12v is likely.
Then test the voltage at each of the bulb fittings.
G4's can become loose because they get damn hot, I assume yours look like this:-
Pull each bulb out, clean the contact wire and bend it out a bit, Then when refitting the bending will act like a spring and hold them in better.
It could be that the transformer is balanced and doesn't like any of the bulbs being blown.
If your has 12 x 10w G4's = 120w and you have run it with a few 'popped'- the transformer then over feeds the remaining bulbs and they will have had a much shorter life due to receiving more power than they like.
The bulbs are 99p each from TLC, you could start with buying 20 (so you have some spares) and replacing all 12 and seeing what happens for there.