There are 2 issues that make running cable in cavities a bad thing.
A building regs one that you should not bridge the cavity in a way likely to lead to the ingress of water. This is a problem if the outer wall leaks, water runs down the cable into the inner wall and spalls the plaster.
This is not relevant if the cavity wall is not like that, or if you can guarantee the cable not to meander from one leaf to the other. Note wall ties are always made with a twist in the middle to drip water before it gets to the dry leaf.
The second is electrical - it is not possible to guarantee that the wall won't be filled with foam or similar, causing the cables to either decay or overheat if fully loaded. For an antenna cable, as the price of failure in service in service is irritation rather than a fire, as it would be with a shower circuit perhaps, I'd ignore the second one.
Usually done with string, washers and magnets, and a lot of patience, oh, and drop the string from the top down - gravity is your friend.
Sometimes at the bottom end, feeding in a big loop of old steel tape measure into the cavity before dropping the string improves the chance of capture.
If you have cavity trays it will be impossible without messing up the wall. Is this really the way you want to do it?