Variants and Vaccines

If the vaccines produce anti-bodies to the virus in question, they will most likely work for a random strain.

When the body naturally produces anti-bodies (because of a previous infection), it makes sense for the body to also protect against some of the future variants which it expects to see... so, same goes for anti-bodies produced because of a vaccine. They'll work against most random strains.

I would be much more worried about the strains that will evolve out of the vaccinated population, which will obviously be immune to the vaccines. Give it a few cycles of this and we'll have a bunch of strains which are immune to a bunch of vaccines, very similar to the flu vaccices, except I think there's going to be way too many strains for coronavirus.

Since coronavirus is way too common in mammals (compared to say, the polio virus) and has a lot of variants and strains, there's no way that the anti-bodies will last long (if they did, our bodies would be full of anti-bodies for everything that exists at all times, making us succeptible to way too many auto-immune disorders). The anti-bodies will last a few months at best (because that's how long the body expects coronavirus pandemic to typically last).