I’m going to offer an alternate view than most of the other posters…
I have struggled with this with my horse. He’s coming 6 and I started him myself and from the beginning he’s never really been the type that can come out of the barn and go forward freely. I thought this was an issue or a hole in his training, (and that is part of it), but it’s also a question of how he’s built. He’s got a high set neck and naturally goes around hollow. The initial years I didn’t worry too much about how forward he was, I just focused on relaxation and getting him to stretch into the hand, and usually, by the time he’d warmed up, once he was stretching and bending nicely, he’d be nicely forward.
Last summer I had a few lessons and clinics with some folks who were in the die hard ‘he needs to be more forward!!’ camp. And I started questioning my previous approach and started thinking he needs to come out and go forward, period. I worried less about relaxation and bending and more about forward. I did all possible variations of everything that everyone suggested above. None of it worked and in fact everything started to get worse. We started a downward spiral and both of us got super frustrated. In an effort to stop the negative spiral I started doing more cavaletti and jumping, and that helped, but it didn’t help with his forwardness on the flat at all.
After about 2 or 3 months of this, I got help from a western friend of mine. She immediately noticed that he was holding and wasn’t truly relaxed or accepting of my lateral aids. She had me return to what I had been doing before - establish relaxation and bending and only THEN ask for more forward. She had me push him quite drastically OVER with my inside aids (making what I wanted crystal clear), then straighten and ask him to go forward. Surprise surprise, once we’d done a bit of the lateral bending, he relaxed, he was loose, and he could go forward. I honestly think it was NOT POSSIBLE for him to go forward UNTIL he was relaxed and nicely soft to my lateral aids. I was trying to kick him forward when he couldn’t physically respond so it’s no wonder it didn’t work.
Now we have moved past that, and he comes out nicely forward but only because he’s relaxed and nicely soft through the body first… if he comes out sticky, I always establish the relaxation/bending first before worrying about forward.
This won’t work for every horse, but it’s what works for us, and it’s the only thing that works for us. My horse has a naturally high head carriage and used to have a big under neck (he was born that way), so for him relaxation both longitudinally and laterally HAS to come first.
Good luck!