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Hi Chuck!
You should also be "MOST perplexed" by the same behavior of moving electric
charges and their electro-magnetic fields:
Suppose I am in my car, parked outside a building containing a huge static
electric charge (perhaps a Van de Graff dome).
If I now depress my car's gas pedal, I will encounter a magnetic field
generated by the (now moving) electric charge. My gas pedal action did not
move the charge, I simply moved into a new frame in which the
electromagnetic field of that charge has a magnetic component.
Bottom Line: Your conundrum is not new or unique to gravi-magnetic
phenomena. We have long experienced it in Maxwellian electrodynamics.