Maybe this is a necro-thread resurrection, but I just bought the game a few days ago and had this bug. So, what I am going to write may seem a bit overly detailed, but it is for the guy who isn't posting that needs to read this.
First, the instructions are VERY basic in the PDF instruction book for the game. You have to use a menu on a solar system to open up the BattleRider Managment screen at the left side of the screen, one of several buttons on the little diagram of your planetary system orbits (building ships is another button). This is provided that system built a few BattleRiders, and you have a fleet in that system that has a cruiser of bigger ship that is a BattleRider carrier.
You go to a screen that is almost as basic as a screen can get. On the right are your carriers, on the left your unused battleriders. You have to dig into the ship menus on the right and select a carrier. Go all the way until you are to a specific ship. When you get ship, then a bunch of slots magically appear in the middle of the screen.
You can drag and drop individual battleriders into slots, but that has been a bit buggy for me. It won't always load the riders. I found that just double clicking on the battleriders on the left auto-loads the ships, and does a better job than drag-and-drop.
Again, the fix I had for stubborn BR's is to double click them to get them on and off a carrier.