If by our goal is to see the fields due to some charge or current, look at Jefimenko's equations. For electric fields electric field they can be caused by charge density, the rate of which charge density changes or by the rate that current density changes.
For magnetic fields they can be caused by current or by the rate at which current changes.
Since you have a wire it doesn't have much charge density (just some along the edges to keep the current flowing in a circle) and if you have current and move it fast then by going fast enough that contribution can dominate the and be the main source of electric field.
Similarly, if you shake the wire fast enough then the change in current can cause more magnetic field than the current itself.
However this has to be pretty strong shaking, at some point the wire might no longer be able to drag the electrons with it. And once the electrons get left behind nothing is making them shake any more.
But what you have now is strong electric and magnetic fields you are actually producing radiation (when you have both) and that is carrying away large amounts of energy and momentum. This radiation is expanding like a ball, expanding at the speed of light and filling that region with equally strong electric and magnetic fields.
You've basically made a radio or television transmitter, based on the frequency you shake it you could find the station or the channel.