In my opinion the EM waves would continue to propagate indefinitely considering literally there is no obstruction to it, because wave itself is basically an energy being propagated in the form of disturbances, and we know that energy can neither be created nor be destroyed, so even the source of EM waves is turned off the waves already produced until that extent do continue to propagate, provided there is no obstruction factor as the energy can never be destroyed according to fundamental laws of Physics.
In fact we observe this to some extent in our daily lives, like if you are on a phone call and suddenly the network drops, you still hear the words spoken after that drop.
Also if you consider light as an EM wave If you are in a closed room and suddenly switch off the tube light, you'll find that although the tube light has been turned off, the light remains for some time, before completely vanishing.
this is just my personal perspective, but I feel it to be valid.
Do let me know what you feel on this