I'd rather suspect a seized starter solenoid - the frees off when cold.
I'd go with that as a first place to look as well.
Obviously check that connections to the starter are tight - you could check with a 12v bulb connected between the solenoid connection and the body that the solenoid is actually getting power when you try to start.
Had the problem on one of my Granadas - it turned out to be some muck in the solenoid itself. Magnetic materials - iron etc lose their ability to be magnetic with increasing temperature until a point is reached where they can't be magnetised at all - the curie point.
http://en.wikipedia.org/wiki/Curie_temperature
In the granny the muck was making it difficult for the solenoid to operate to the point where it was
just capable of operating when the solenoid was cold - but when the solenoid was hot from the general heat of the engine the reduced magnetic effect was enough to prevent it working. I got it working once - having figured out what might be going on - stuck at the shops unable to start - I went back to the shops and bought a bottle of water from the fridge - poured it over the solenoid - and it started ok for me to get home.
The solenoid is only rated for intermittant operation so when you keep trying to start without success it gets even hotter and makes the situation worse.
You could isolate this problem by heating the solenoid with a hair dryer and then trying to start.
To fix, remove the starter motor and dismantle the solenoid (don't know how easy or not this is on a BX - there's different starter types anyway). Clean it thoroughly and reassemble it completely dry - no lubricants and no muck. Make sure that any bolts holding the magnetic circuit together (the iron around the coil etc) are as tight as you can reasonably get them (any muck or a gap in the magnetic circuit will reduce its effectiveness quite a bit).
Make sure everything is clean on reassembly and have a look at the brushes while you have the starter motor off.
restarting when the engine is hot. After 30-40 mins when the engine has cooled it'll start again.What can cause this? 
as always. Panny
I thought the auto box's inhibitor switch simply open-circuited the starter energiser feed.
