How deep does a gravity well need to be to remove particles from a planetary body? I almost considered asking this question on WorldBuilding, however I wanted the brute mathematics on the subject, so please excuse the theoretical nature of this question.
I understand the basic nature of gravity wells, in that they are analogous to literal wells in the fabric of space. I also understand how bigger bodies in space have larger gravity wells, how big would that gravity well have to be to dominate the gravitational attraction of objects within it's well? How big would it need to be to give a object exit velocity from a planetary body? I understand that the gravity well and the planetary body would attract, but is it possible to move objects away from larger gravity wells with a smaller, but closer gravity well? My thoughts are that if you are close enough to the well, you would receive much more attraction than if you were farther away. Is this correct? 
While an apple does have gravitational attraction, you do not feel any noticeable pull from the apple. What's the minimum size that a body must be for you to notice gravitational pull? What if the hill spheres were aligned in such a way that merely jumping would allow you to change gravitational sphere, à la Mario Galaxy?
In my crude drawing I've illustrated my theoretical situation. The object (the dwarf) is close enough to the moon to be within its dominant gravitational pull, and is therefore being pulled away from Earth.

John Rennie pointed out the concept of Hill spheres in a comment, which might be useful in trying to figure out if they could be used to dominate nearby objects.
 A: Assuming that the gravity well acts as an  attracting  mass whose force follows Newton's law of gravitation, your well will suck the object from Earth's surface when the force from the well is larger than the force from the earth. That is, 
$GM_{earth}m/R_{earth}^2<GM_{well}m/r_{well}^2$, 
where $r_{well}$ is the distance between object and well. Earth will be affected but less, because it is at a larger distance $R_{earth}+r_{well}$. Thus, Earth will be affected less when $R_{earth}>>r_{well}$, that is when the well is close to earth.
The above condition (for the well to work) can be simplified to: 
$M_{well}/r_{well}^2>M_{earth}/R_{earth}^2$ 
A: What you need to pull an object off the surface of another, gravitationally, is for the tidal acceleration $F_{\rm tidal}$ from the "external" body at the surface of the Earth (or whatever other body) to exceed the gravitational acceleration $g$ of Earth on the surface. This comes with the caveat that if the tidal forces are too strong, they will start to break up the Earth, and it won't be so much pulling objects off of the surface as pulling the surface off with objects still on it...
The approximate condition, which holds if $R$ (the distance between the centre of Earth and the centre of the "external" body) is much larger than $R_\oplus$, the radius of the Earth, is:
$$\frac{2GM_{\rm ext}R_\oplus}{R^3} > g$$
This is actually just a different take on the condition given by Wolphram jonny, which you can see if you set $g=GM_\oplus/R_\oplus^2$.
