This is a apparently simple matter if one uses a xover without worrying about what it does or how the parts behave and interact with the amp, but if one wants learn in detail the behavior of an xover is a complex subject.
In the lasts years of US Audio mag they published a reactance/inductance/impedance graphic when testing a speaker that allowed the reader see some speakers are not a sweet peach to driver even being rated easy to drive by the manufacturer.
The fact is that xovers consume low level musical signal/energy (mV/uV) to work, unfortunately this energy is lost as heat.
Specifically, if I put a high pass filter on a speaker because a sub will take those lower frequencies, I understand the lower signals will not make it to the driver, but does it take the filtered signal and dissipate it as heat or instead cause something 'different' to happen in the circuit with the amp.
A capacitor mods impedance.
A air core inductor att the freq 6dB/octave, the most destructive inductors of the musical signal are the iron cores, but they are cheap and attenuate 18dB/octave if I remember correctly.
The pragmatic reason I am asking is understanding exactly how it affects the demands on the amp. I.e. did it lighten the load on the amp allowing low wattage amps to drive the frequencies that made it past the filter to greater levels, or is the power wasted?
I dont see how the load to the amp would be relieved, the xover parts add an additional load to the amp drive and lower the sensitivity of the speaker in question, the lightest load an amp can have is to drive the speaker directly, without xover IMO. This is a multi-faceted subject that allows one to have multiple point of views, your questions are very wide broad which exceeds my little knowledge.