Yes, we bought our pony primarily for gymkhana, even though he has a jumping background. Again, the less stressed she is when she introduces him to new games, the more he just takes all the new equipment in his stride (he had never done gymkhana before). Watching her vault on him is amazing 
It also helps when she sees others hop on him and make him do stuff she finds hard, my friend points out why they can do this and she listens and takes it on board.
I just wonder if your DD is slightly stressed with the situation and that's transferring to the pony.
I also figure pony's can be naughty! Our friends at the farm specifically buy 'challenging' ones for pennies (almost!), then working with them to make them into fab jumpers or games ponies. Two weeks ago, we took one of their mares to a jumping course, she reared and reared and reared and refused. Our friends DD (who is 11) just laughed, stayed on her and figured how to get her over the jumps and made her do half the course. But that takes guts and experience 
Problem is, my DD thinks she needs to be like that, she can't - she's only ridden properly for less than a year (although hacked all her life). When she's with the girls, whose ponies a fast and furious, you can she her trying to keep up and then the stress keeps in. When we are out with our other friend, just with her DS and his pony, my DD and consequently the pony are completely different.
Apologies for the typos previously, I was writing and dashing!
Oh - and we don't pay livery as it's a family farm but we have to do everything, the whole shebang. DD got top marks for stable management and tack cleaning at PC camp, some of her friends with 'amazing' ponies didn't even know how to muck out. I am proud of DD for realising the work load she was taking on when we got the pony.