Suggestions on encouraging use of hind-quarters

Hey, I'm on the lookout for exercises. I've been working on one of my trainer's project horses for about six months. She's a seven-year-old paint, over sixteen hands high, and I'm training her just to generally be a good horse, but I also show her in dressage, and some little over-fences classes. She's basically well-built, uphill, sturdy legs, acceptable neck and all of that good stuff. She's going well forward, which was a struggle to start with, and she understands bit pressure and is pretty light to those aids. She tends to be a slow-moving horse, but again, I've worked extensively on making her move out. I've done a lot of work to get her shoulders upright and body straight, and lots of untracking, shoulders-in, half-pass-ish movements, that sort of thing to get her more flexible. I feel like I've run into a rut I see with a lot of riders and horses, focussing on what's in front of me and forgetting the motor. Her front half is going well but we don't have enough drive from the hind end. Her lack of back-end use is very evident at the canter, where she can be soft on the bit although her gait is generally sloppy and lacking propulsion. Long story short, I was wondering if anyone had any advice for getting her to really use her hind-quarters to propel her forward.

Transitions constantly help get the hind underneath. Hill work helps build up muscle though if she is lacking. I incorporare long low hils in my road work as they are tougher then steep quick ones

Definitely transitions, particularly trot-canter-trot. Lateral exercises are good as well.
You said you're concentrating more on whats in front than behind. So don't. Concentrate on feeling her hind legs and ignore her forehand! You can ride all the exercises in the world but if you're worried about having her head down those hind legs aint going anywhere!

I have a training horse that has the same issue. We are now doing a lot of caveletti, ground pole work and backing patterns.

On the long wall, work on two tracking and roll backs.

Leg yields, transitions, flexing, as well as shoulder ins and pirouettes are very effective exercises for getting a horse off the forehand.

Rollbacks help out a lot, and also what I like to do is when I'm out riding I just sit down and stop randomly. At any gait. I then back him up a couple steps and let him sit there for a couple seconds. It makes him pay attention to me more as well as getting him on his hindquarters.

Yes! If you access to any fields or open areas, preferably a little uneven in terrain, take your horse out there and trot around on the bit, not all strung out. It will build up so much muscle and increase his balance so well in such a short period of time, it will knock your socks off. I know this because last week was very humid in my area, so my horse spent all last week "playing" in a shady back field. We spent 6 days just trotting and cantering around like idiots. Yesterday it cooled off, we did our first ring work day in over a week, and the big, forward, fantastic trot that came out of my formerly polky puppy was incredible. Love transistions too, but trotting on uneven ground freakin' rocks!!

To add to the other ideas, I do a lot of small circles at a trot and a canter. I think circles pretty much force them to use their back and their hind quarters effectively because there is so much balance involved. I would just make sure you do even amounts on each side.

As my coach says, you can do exercises until you are blue in the face, but nothing is going to change until you change how you ride.
I highly suggest getting some dressage lessons from a very knowledgeable instructor in your area and really work to get you programmed to get the horse moving more from behind. Otherwise yes, transitions DONE CORRECTLY will aid engagement. I always warm up my horses with transitions from gait to gait and later in the ride begin to use transitions within the gait to further increse difficulty. They must however, be done perfectly every time for any result to be seen.

Good luck!

Very, very helpful, everyone. I'll be doing transition work, then, and continue our work on flexibility and straightness. I'm certainly intending on getting dressage lessons, too. Thank so much for all your suggestions!