[firebacon]

Does the chance of seeing a vaccine-defeating mutation increase after injecting parts of the population with a less-effective vaccine (compared to the baseline scenario of no vaccinations at all)? What mechanism is responsible for that?

The answer is only obvious to me if both the original virus variant and the mutation compete for some kind of shared resource. But that shouldn't be the case here, right?

[mlyle]

The original virus and the mutation compete for survival in a host body with a partially effective vaccination.

If you generate only a subset of the typical complement of antibodies that a vaccinated individual does, and are less protected and become infected as a result... then any virus variants that emerge within your body that escape any of your antibodies will have a fitness advantage. In turn, that virus will have an advantage spreading to other vaccinated individuals, too.

It is a low probability event in each individual, but if you generate enough individuals like this and infect them all, it's sure to happen eventually. We can't really estimate what the probability of this happening is, but it's certainly less likely if the efficacy is high and there's less disease circulating.