Our ops manual says that the provisions of the MEL cease to apply at the point of despatch - and then goes on to quote Article 129(2)(a) of the ANO which defines "the point of despatch" as the time at which the aircraft first moves under its own power for the purpose of commencing a flight.
So if a problem occurs whilst you are pushing back or starting engines you return to the gate. If it occurs whilst taxying you can excersise your commanders judgement (for which you are being paid huge sums of money) to decide whether to return to the gate or continue the flight - if you deem it safe to do so. On the occasions that it has happened to me, I have consulted the MEL and if that document has said that the engineers can raise an ADD, and if the weather conditions etc are all suitable to carry the fault if that was done, then I have continued and snagged it at the next station.
Admittedly, our flights are around Europe and short sectors. If you are off across the pond you might decide differently.