@melanol Replays use their own copy of the game files, so even if the replay you're watching is of the game you just played, the client will need to download the game files for the version of the game that the replay is for.
While you can't stop this, you can mitigate some of downloads by enabling "Game data cache" in client > top left menu > Settings > Data and caches. This will tell the client to keep a copy of the game files for the replays you watch so that if you watch another replay with that same version of the game files, it doesn't need to download them again. I would suggest that you review the "Game data cache validity in days" setting to ensure that the client will keep those replay game files for as long as you'd like (use 0 for no time limit).