Coil current changes while working
Interesting question.

If you're exciting the coil with DC, and the plunger is free and not at the center of the coil, then as the plunger is pulled into the tube, the inductance of the coil will increase which will limit the instantaneous current that can be supplied to the coil by the driving constant voltage source, which will limit the speed at which the plunger moves toward the center of the magnetic field.

Also, because of the mass of the plunger - unless you've critically damped its energy against the friction between the plunger and the ID of the tube - it'll oscillate for a while before it settles down at - more or less - the magnetic field's sweet spot.

If you exercise the coil with AC it gets more complicated. Do you want to go there?