Bud grafts on fruit trees stopped growing.

I was wondering if this is anything of concern. i bought a 5-in-1 dwarf apple tree (shown in photo), and also a dwarf 5-in-1 stone fruit tree (they called it a fruit cocktail tree). I have had some growth for sure from some of the bud grafting, especially since the apple tree is 2 years old now. But i have noticed that on both of them, some of the bud grafts havent grown yet (although they dont exactly look dead). And on both of them, there is one graft that sprouted only a couple of little leaves and then stopped. anything that i can do to encourage those remaining grafts to get going?

I would suggest root stimulant - you should be able to get liquid form at a garden center. Root hormones tell the buds to grow, and bud hormones tell the roots to grow. Poor top growth on transplanted trees usually indicates that the roots haven't established.
After the initial application, would apply I would apply a dilute solution every two weeks.

Thanks for the advice. The stone fruit tree is indeed a brand new purchase for this year, and the apple tree had a really rough first year. I first had it planted in the bottom of the yard thinking that was a good spot for it. Well apparently the deer also thought this was a good spot for it, lol. I had to literally dig it back out of the ground and move it into that giant pot to keep them from totally killing it. I waited until a rainy day when the ground was wet to get it out and thankfully i would say i got around 95% of the root system out intact, i call it dumb luck. They shouldnt be able to get to the other tree because that is fenced off.

Here are some closer up photos that i took this morning. I am noticing some of those grafted buds on the apple tree turning a more red shade of color like they want to try to wake up and get going. also please point out if there is anything i need to remove, as i am a newbie at growing these

General rule is do not prune until roots are established. However, if you are worried about your grafts and have something you don't need near enough to a graft to be competing with it, you might want to prune it back OR bend it down from upright to between 45 degrees and horizontal - you might not be able to do this all at once without breaking it - work in increments. Also, some of your grafts looked like maybe the growing twig and the stunted bud were the same graft? In which case if you have the twig, maybe the bud doesn't matter so much?

on the apple tree, if you are speaking about the graft in the first photo concerning the "twig". that was one graft that took off the first year, but the deer munched it down to about 5 inches (which is just out of the photo shot), but that has started bouncing back this year.

The second photo is that graft that grew just a couple tiny leaves then stopped. that branch that is coming out just below it i actually believe is a branch coming out of the stock itself, but i didnt want to risk pruning that off and stressing the plant so much that it died since i already had to remove it from the ground and into the pot, i wanted it to at least get acclimated and a little new growth before the winter hit. Now that we are well into spring, should that stock branch come off, perhaps that will stimulate that bud to get going?

That is a definite maybe. If there is other healthy new growth on the tree, that branch could probably come off. If you can cut it just outside the branch collar at the base without damaging the other bud or bark below bud, that would be best. If not, cut it off a little farther out & rub out any buds that break dormancy between the cut and the base.

oh yes this first photo that i posted above is what has grown out so far this year.

just thought i would give an update. I have done several things. one thing is that i did remove that stock branch that was just directly underneath one of the grafts. I also sprayed the tree with a micro-nutrient foliage spray. and lastly i did something else that i had read online, i lightly sliced into the bark of the tree about 1/2 inch above each graft point (something to do with altering the flow of its chemicals so that the buds get the right ones they need to break dormancy.). and a couple of weeks later, i am seeing some action on 3 of the 4 dormant grafts, including that one that started putting out a couple of leaves and then stopped. no change yet for the stone fruit tree, maybe i will have some luck later on in the year

Good to know - thanks for the update. I have seen photos of the slicing above the bud - it looked to me like two tiny short parallel slices to make a little gap like the tip of the blade of a flat-head screw-driver. I don't know how much difference that makes, or if the single slice heals too quickly or not.