That's not how it works. If the planet is N light years away from earth, the transmissions you receive would be those sent 20000-n years ago. along the way you'd get most of the messages sent during the 20k years as very red-shifted light, and extremely rapidly from your pov.
I think you may have that wrong (If I'm understanding you correctly).
If you traveled at the speed of light then you'd be at the planet in 0 time your time, and you would arrive with the first of the broadcasts over those 20k years. So once on the planet you'd get to watch all 20k years of broadcasts.
If you traveled at 1/2 the speed of light (not focusing on your dilation for the moment), then you'd still beat 50% of the transmissions and have 10k earth years of broadcasts to watch.
I think the question is what % of the speed of light is a gamma factor of 20k/10 = 2000. That's something like 99.9999999% of the speed of light.
Meaning you would get there before 99.99999% of the broadcasts had arrived and you'd be able to watch just about all of them in real time over the next (just less than) 20k years.
Assuming those two planets are roughly in the same reference frame, the only way 20k years could pass on earth w.r.t. your arrival is if the destination planet is 20k light years away.
If the destination planet is moving relativistically away from earth at an appreciable percentage of the speed of light then I couldn't say what the math would be. Maybe still the same, maybe not.
