If an engine cuts out, it's usual practice to check if there's a spark and if fuel is reaching the carb.
The manner in which it cuts out can also give a clue - cuts out suddenly is likely to be ignition, stumbles as bit, won't rev up then cuts out is more likely a fuel supply issue. Also with an ignition problem, continued attempts at starting will likely cause the spark plugs to become wet with unburned fuel (because you're missing the sparks).
To check for a spark, pull the HT lead that goes from the coil to the distributor cap at the distributor cap end and hold the end close (approx. 5mm) to a good earth on the engine (using a pair of pliers if you nervous), spin the engine on the starter and check for a good spark. Some people pull one of the plug leads and hold it close to an earth point to check for a spark, but my way gives you 4 times as many sparks and removes faulty plug HT leads/fau;ty distributor cap/faulty rotor arm from the equation. Be aware that some of the older Fiats had a resistor in the rotor arm that could fail...
As others have said above, various ignition components can work when cold then fail when hot, many of the parts mentioned are worth renewing on an older vehicle anyway to ensure reliability.
Be aware that modern aftermarket replacement parts are not always good quality e.g. some ignition condensers have been known to fail shortly after renewal, some were even faulty new, out of the box, so best buy a good brand. It might also be a good idea to carry a spare if you're contemplating any long trips in your campervan. (there has been lots of discussion on ignition condensers over on the 500 Classic section of this forum).
Also, a failing condenser can cause the contact breaker points inside the distributor to become darkened/burned in appearance, so these should be renewed also if so damaged.
To check for fuel reaching the carburettor, carefully disconnect the fuel pipe at the carb. By carefully, I mean hold a rag in place to catch any spillage - you don't want to spray fuel near a hot engine...
Another problem that can affect some rear-engined Fiats is fuel vaporisation - this is where the fuel in the pipes between the fuel tank and fuel pump or fuel pump and carb becomes too hot and turns to vapor, resulting in lack of fuel reaching the carb. Should this turn out to be a problem, I'd suggest doing a search over in the 500 Classic section of this forum for possible cures.