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In Edit Mode we have Proportional Editing mode. I need something similar in Object Mode. I need to edit properties of several Objects (scale, position, material diffuse color), but the value dependant on distance from some center point in 3d space, similar to Proportional Editing.

Say you have meshes, for the sake of simplicity just cubes, in a grid. Now you want to pull them closer to one of the cubes proportionally, which is what would happen if you selected a vertex, edge or face in Edit Mode and started scaling. How can you do that? Or I want to scale the cubes each along their center, which is possible with setting the Pivor Point to Individual Origins, but change the center and the result dynamically (for an animation).

About my specific use case and what I need this for: I need to create a simulation on how human vision works, the part about seeing more "pixels" in the center of our vision relative to the edges. The animation consists of a 160x120 grid of planes and there is an eyeball looking at the plane. The eyeball mesh rotation is locked to an Empty on the grid of planes. I need to be able to, as I move the empty, have the objects (Planes) closer to it be pulled closer to each other in a smooth falloff and since they get so close also scale them down so they won't intersect, and also scale up the planes farther away to reduce the gap created between them. I also need to have the planes pulled closer to get their material diffuse color to red and go blue farther away. This is for an animation so the empty is going to be constantly moving, so changing which planes it influences and how much.

Does not cnecessarily need to be done with meshes and maybe particles would allow such a feature, so would work for my specific use case with simple identical planes. But for other use cases where such Proportional Editing-type animation is required on more unique group of meshes the answer might not apply.

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What you describe is similar to Animation Nodes.

This plugin allow to manipulate with objects based on algorithm, described by nodes.

enter image description here

Unfortunately, Animation Nodes is to difficult to explain in one post. But I give you a file, there you can start with.

(Note, you need to install Animation Nodes to view this file)

This is the nodes, that I used in the scene:

enter image description here

Here for every one object in my group (all cubes are in this group) it mix Initial transform and position of empty and set result to position of cubes.

Initial transform sets in T menu, here (select all objects and press From current transform):

enter image description here

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