The simplest. most portable and least expensive thing I've used is surprisingly effective but requires a little sweat equity. All of the parts can be had from Home Depot. It takes longer to describe than to set up!
The most difficult thing is to come up with four stout posts, poles or other anchor points on the range. Some 4x4's buried in some post holes works well, one at each corner of a pit, say. Or four large portable barricades with big bases that won't tip over.
Between the two downrange posts stretch a guy wire (cable). Use whatever hardware you find convenient to attach it to the posts. A turnbuckle or come-along can be used to tighten it as required, but it doesn't need to be super-tight. Place a couple of pulleys or sheaves to ride on the cable (the kind that won't fall off) prior to securing the second end. Attach a a pulley to each post with a short length of rope (just loop it around the post/cable junction) so that each these pulleys hangs about 6" or so below the cable (the short rope will allow these pulleys to have some freedom of movement--this is very important).
Build a simple rectangular target carrier out of 1x2 strapping. Hang it from the pulleys that ride on the cable.
At the uprange posts or anchor points attach pulleys in a manner similar to that done at the downrange posts. Again, attaching these pulleys in a manner that leaves them free to twist, e.g. on a short length of rope, so that the operating rope has a fair lead through them is very important for smooth operation.
Finally, get a very long piece of rope. Route it in a big loop: attach one end to the left target carrier pulley, then route it through the left downrange anchor pulley, then the left uprange anchor pulley, then the right uprange anchor pulley, the right downrange anchor pulley, and finally attach the other end to the right target carrier pulley. If you have extra rope coil it neatly at the center of the uprange end of the rope and tie the coil off there, or just cut it off to obtain the custom length loop for that range.
Use good quality rope, pulleys and hardware so that operation is smooth.
Operation is simplicity itself. The target operator grabs the rope at the uprange end and runs back and forth behind the shooter to animate the target laterally in whatever manner is desired. A bit tiring, but complex motions and target behaviors in response to shooter actions can be easily accomplished. Cover and concealment can be set up in front of the moving target plane to add complexity.
This system is super simple, flexible and can be engineered to be completely portable. Downsides are the human power factor, lack of repeatability (but this is training, not competition) and target motion can be telegraphed by sensing operator motion or motion of the apparatus. Still, all-in-all very challenging.
aa