It's a little complicated. The probability of a mutation per-transmission is roughly constant, and the rate at which new variants (of any kind) appear is roughly proportional to the rate of new infections. So the more rapidly the virus spreads, the more mutations we expect to see. But another effect is that the amount of pressure for variants with certain properties, like being more transmissible, also depends on what we do. For example, social distancing slows the rate of spread, but can increase the rate at which the more transmissible variants outcompete the others. A mostly vaccinated and/or recovered society will likewise mean the rate of spread is very low and the impacts are largely mitigated, but there will exist a stronger selective pressure for variants that can evade prior immunity.
Another factor to consider is that variants may diverge from one another more if they are transmitted in independent groups. That could be because they dominate in different geographic regions (like northern vs. southern hemisphere), or that one spreads more in children than adults (which we see a lot of now with most adults being vaccinated and kids being back in school). Basically, just think about and apply all the rules of evolution, and you'll have a good understanding of how and why a virus changes over time.
Looking forward, a real possibility is that we end up with variants that are sufficiently different from one another that each vaccine is quite effective against some variants but not so much against others. That is essentially what already happens with the flu. Each year, the vaccine manufacturers try to predict next season's flu behavior and tweak the vaccines accordingly. But they can get it wrong, or the flu variant that dominates in a region may be a different one than expected (there are usually two serious flu variants spreading in each season, named A and B). This is part of what limits flu vaccine efficacy every year.