Modern Vespa

I have a Piaggio MP3 250 and when it's warm and I turn it off, when I try to start it again, it starts but then stalls shortly after. When it cools down, it starts normally. Does anyone have any idea what might be causing this? Thank you."

Fuel pump.

The fuel pump is checked. It's OK they said. Thanks for the answer.

Check your VIN number for the dodgy fuel pump recall

when were the valves last checked / adjsuted?

At least for Europe, the Piaggio recall form does not list all recalls!

https://www.vespa.com/en_EN/recall-campaign/

Only the dealer or Piaggio itself can provide precise information.

The problem starts in the summer. I think something is getting hot. When it's cold the engine works as nothing happened.

The known thermal problems with the fuel pump can be easily diagnosed: If the fuel pump does not whir when the ignition is switched on when hot, but does so again when the engine is first started or after it has cooled down, then 99.9% of the time it is the fuel pump...

In other words: the engine can only start if the fuel pump hums briefly when the ignition is switched on and a corresponding fuel pressure has been built up.

If the fuel pump hums and starting is still not possible, the problem is somewhere else.

Did 'they' take it for a test ride to get it good and hot, then try to restart? They may have just checked the recall database, and noted that the pump pressurised just fine when cold.

Next time it's warm and you stop, listen for the fuel pump 'whirr' for two seconds when you first switch on and before you try to start.