During mesh deformation, if a composition on the boundary changes, undershoot/overshoot can occur. A good example of this is hill_diffusion_field_boundary_conditions.prm (layer 1). The Layer 1 composition of the top boundary is determined by the function if(t==0,1,0) if new material is added to the domain through that boundary. In the images below, cyan = 0, blue is negative.
Timestep 0

End time (new)

It would be good to modify this test, and any other related tests, with a parameter / parameters that reduce/remove the overshoot.
(This problem is not fixed by #6829, and is present with AMG and GMG)