Do the SUVAT equations work for acceleration that changes with time?

It depends on how much it changes. For instance, if some object does undergo as good as makes no difference constant acceleration, then the SUVAT equations apply quite well to the circumstances at play. If there is a non-negligeable time dependency made of the acceleration by the phenomena at play, then the strength of the SUVAT equations is greatly reduced as they are derived from assuming that if an object is in motion, the acceleration of that object is uniform. When the acceleration is indisputably non-uniform then the results drawn from assuming the uniformity that isn't there will be greatly deviated from reality.