So, something from the sub-Reddit:

EY: If you throw a Transfigured object and de-Transfigure it, it continues moving at exactly the same velocity, regardless of momentum and kinetic energy. There are at least two good reasons for this, one of which is that, even in MoR, Special Relativity is simply the way reality is; you can add laws that violate conservation of momentum and even conservation of energy (though the latter requires the insanity of single-world QM, but then so does the Time-Turner), but in a Minkowskian universe the idea of a privileged frame of reference is too incoherent even for magic. The other reason is that if you didn't know about conservation of momentum or energy, of course you'd expect the object to continue with the same velocity.

My limited physics knowledge is failing me. Assuming that you stick with relativistic velocity transformations, why is conserving momentum problematic?