Sorry for not being around here recently. In the ququart case it would take more time per j but there's also less j required. Either way, it's still scalable. If you have a more general way to implement controlled unitary gates then it is possible to make the circuit for multiplying that number with just knowledge of the N that is the mod and the value without significant classical overhead.

In the case of N=15 with d=4, we might be able to find the period from the truth table, but that is just inherent characteristics, correct?

Yep.

Any reference about the circuit designing in the case of qudit?