Hi Jess. I’m by no means an expert but I have found that there are some types of shadows and techniques that are going to give me more or less shadow “fallout”.
Pressed powders seem the most fragile and prone to fallout to me. So I am really careful to tap out the application brush before adding it to my lids, add shadow lightly & slowly, blend across the eyelid as I apply it (add a little shadow, blend add a little more shadow, blend, etc.), and don’t overload my crease too much (I have hooded eyes so I blend above the crease anyway).
Creamy shadows are a lot more forgiving is seems, and while I haven’t used a baked shadow, I imagine they would be a bit better too (I have used some baked blush/highlight/bronzer and noticed they were pretty keen on staying where applied.)
As mentioned by others, an eyelid primer will not only cut down on fallout, but keep your shadow brighter and longer lasting (I am guessing because the shadow isn’t falling off as much). If you don’t have some primer, try a little application of concealer on your lid before adding shadow.
Besides being slow and steady in your application, I have found that a simple index card in your makeup kit can be really useful in not only guiding the angle of where the shadow (or eyeliner) goes, but also catching a bit of fallout. But it can git a bit difficult to do everything at once.