Does the problem happen at all settings of the 12-step power control switch? During the short time that the welder does work, does it have a power level that is appropriate to the switch setting?
The fault could be due to a poor connection which creates a high resistance. During welding, the flow of current produces heat, the resistance gets higher, and the problem will get worse.
The bad connection could be external - i.e. the Eurotorch, or the return cable, Dinse plug, or work clamp. You could test with alternative parts. It could be internal, which would mean opening the casing and examining all connections in the secondary high-current welding path. Look for loose, corroded, or burnt connections. If in doubt, open the connection, clean, and re-make.
There could be a connection fault on the transformer primary ( mains ) side, but that is less likely. That would be the power selector switch, or the contactor ( heavy-duty relay ).
An electronic warm-up fault is possible. One or more diodes in the rectifier could be going open-circuit when it is hot. Repair would probably involve replacement of the rectifier.
It may be possible to get a better idea of what is wrong by measuring the welding voltage between the torch tip and the work clamp. Turn off the gas, release the pressure roller, then measure the voltage with the trigger pressed. It should start about 20v DC, and rise smoothly with each step of the power range switch, up to about 40v DC. Then run the welder till it gets hot and the fault appears, and check the tip voltage again.
You may be planning to check out the fault first, or to buy the welder as-is for a reduced price. In the latter case, I'm sure you can get lots of advice here. It is a traditional transformer-based machine, there's not a lot to go wrong, and it should be repairable - but that is not guaranteed!