Apart from speed, do VSD's also alter current?
VSD stands for variable speed (DC or AC ) drive.
A VFD is a DC controlled Variable Frequency AC Drive.

Modern VFD's are highly programmable to sense RPM, current and control V/f with PWM. There may be a hundred control options or parameters or a simple user interface.

Current is a result of dependent on controlled acceleration, external load factor and magnetic voltage-dependent excitation currents.

With no load, current may be limited by a constant acceleration rate of increasing f while keeping V/f constant.  With a load, it depends how it changes from start to finish, so each application may have custom start-stop profiles.

Thus the acceleration rate can be constant to max RPM or controlled by load or limited by either.   The reciprocal function of braking is similar.

DC motors with full voltage applied can draw up to 10x the rated max current and power with the start surge or full stop with 0V.  Doing this often can lead to thermal issues.