only if you remained the same distance from the earth!
No. Even if they accelerated to relativistic speeds, even reaching the speed of light. The transmission will continue in normal time for the viewer. It’s not going to just slow down or speed up because they are going the speed of light.
The transmission is being observed INSIDE their craft. Which is recording and sending that transmission. The transmission isn’t accelerating away. It’s travelling at its normal speed. The transmission would just be delayed the further they got from Earth.