I am confused. Do I need to switch my fridge to 12v using the switch on the fridge (next to the 240V one) before towing? Or does it cool automatically when connected to the car through the 13 pin plug and the engine is running? My Triton has a separate 12V key switch to turn it on when off grid camping. I have read the manual, but am still confused. It's a Dometic RM 4230/1

In my 2007 Triton 430 I never turn off the red (12Vdc) switch or the green (230Vac) switch. If I'm towing the fridge runs on the 12Vdc element powered by the towing vehicle's supply, if I'm connected to EHU then the 230Vac element in the fridge comes on. If I'm off grid I light the gas.
The three heat sources are not connected in any way and do not run down the leisure battery if left switched on.

The 12V will only work if the 12V switch on the fridge is turned on and the van is connected to a tow vehicle with it's engine running. The 12V power comes from the tow vehicle's alternator, not from a battery.
The 12V circuit that powers the fridge is totally separate from the 12V circuits that power the lighting and pumps. As such you can turn off the 12V masterswitch and if towing, the fridge will still be powered.
There is still need for caution with cars with so called smart alternators. They are programmed to cut output once the car battery is fully charged. If the towbar electrics are model specific they will ensure the alternator keeps producing current even when the car battery is fully charged. This keeps current flowing to the fridge and van battery charger. Cheaper generic towbar wiring doesn't control the alternator and the current to the van can drop out once the car battery is charged.

Brilliant, thanks for that advice. I shall remember to switch the fridge red switch on next time we travel. If stopping for long periods (on a ferry, for example), do you need to switch it off again? Or does the fridge stop working when the engine's off? Don't want to come back to a flat car battery...

We always switch our fridge off - the red switch when we stop off at Services or on the ferry or on the Euro Tunnel just to be on the safe side.

a long long long time ago there was an issue with a wire put in wrong hole in towing hitch electrics that wire had to be moved or bridged. sorry too long ago to remember colour and hole placement a search of interweb will bring up discussions..

however if 12volt switch selected, the lights on, may be the 12volt element its self, think you need a service person to check the fridge.

who was it who took fridge out, put upside down, then after a time put back in right way up, it worked.

(another tale of old there was also the puck owner tale of opening side window finding was stuck open due to closeness of the wall ended up removing the stays. to close.)