Morris Minor Owners Club

had a real pain in the rear with the drivers door not staying shut, thought after much effort it was fixed by relocating the slam plate - one of the screws snapped off corroded etc, was a complete nightmare.
Got it back together where i thought was the correct position - it stayed shut for a whole mile
Anyway back to the drawing board - and found a fix that works.
On the drivers door undid completely the screw nearest the interior of the car, slackened the outer one a couple of turns.
found the ideal thing - some rubber backed slightly conical (so they compress) washers from some tec screw fixings i found lying about. Just one of those seemed the perfect thickness - maybe 3/16 - sandwiched between the door pillar and the plate with the screw going through the washer hole.
Anyhow it really works so thought i'd share, no more slamming the door (good job too as thought it was going to disintegrate some day).
Got quite a few of them if anyone wants to try, it was getting darned annoying

On one car I worked on I had to pack up the catch plate (due to a poor non-DIY rebuilt A post). This gave more engagement between the door catch and the catch plate.

On the pick up I have a similar problem to your Traveller - the drivers door will uncatch itself during driving. I'm tempted to try improving the door alignment (lack of) and also to fit a new door rubber. For the catch to work well, some 'compliance' is needed - ie the door catch needs to travel beyond the catch plate and be 'sprung' back onto the catch to hold the door tight.
My door rubber offers little or no compliance as the door fits badly and the door rubber is shot where it does actually touch

Well dont think that the trav door is any worse than yours for poor fit - closed its actually sat on filler both on the sill and under lower door edge thanks to previous owner.
Maybe i just got lucky but its fine now - did regrind the and of the brass catchbolt too to give a nice flat surface.

A quick check to see how much 'grab' the lock bar is making with the striker plate. Close the door from the outside, then gently pull the exterior door handle and see how much slack movement there is, before the handle actually starts to move the lock bar. With a perfect fitting door, there should be no slack at the exterior handle.

If packing is needed on the lock plate, fibre washers are best, as they provide more grip.

If the door closes securely when stationary but opens when moving it usually means that there is too much flex in the body and some attention is needed. One of the standard tests for when buying a Minor is to drive along whilst feeling the movement between the rear edge of the door and the B pillar.

If the door closes securely when stationary

I'd say mine closes 'barely'