I don't know what you think can happen. Take for example curves of genus at least two or some other class of varieties with a moduli space. Restrict yourself to the Zariski open where the varieties have no non-trivial automorphisms. Then the field of moduli is the field of definition, so a variety is defined over $\mathbb{Q}$ if and only if the corresponding point on moduli is. So you might as well be asking, given a variety defined over $\mathbb{Q}$, how do I distinguish its $\mathbb{Q}$-points among its $\mathbb{Q}_p$-points? Why would you do anything else other than looking at the coordinates? Now, if instead you look at adelic points, then there are other things you can do which may be in the direction you want, like descent obstructions and so on. Being a moduli space may help prove something but, ultimately, it's about points on a variety.