Can the equations of kinematics be used in a situation in which the acceleration varies in time

Yes but it gets a little more complicated unfortunately. The kinematic equations work more easily when you calculate the average acceleration. If that is not possible, you will need to separate the the work into parts where the acceleration is constant. IF you have a problem where the change in acceleration is exponential you would need an integral calculation.

HOWEVER, Even if you are under the impression that acceleration is changing in some non-linear fashion, you are probably over thinking it a bit. Simply calculate the average acceleration and work with that.

If you think about it, acelleration is just an expression of the rate of change of velocity. If that rate of change is its self changing, then that just equates to a different rate of change in velocity, i.e a different acceleration (or average acceleration). SO just take the 'average' acceleration, knowing that there is no such thing physically.