Suppose one sprinkles points uniformly at random on the infinite Euclidean plane,
with some density $\rho$ per unit area.
View the points as disks of radius zero.
Now the radii $r$ of all disks grows simultaneously at the same (slow) rate,
with the disks pushing one another when they come into contact,
with never two disk interiors overlapping.
Each disk-disk contact pushes both disks apart along the line connecting
their centers.
More complex clusters move the disks by summing the disk-disk force-separation
vectors, although I want the disks to move only when pushed, i.e.,
they never gain momentum to drift without pushing.
Three time-snapshots for a region containing $8$ points
might look like this (just a hand simulation):

Here the disks do not yet touch at $t=1$ (blue), but at $t=2$ (tan),
$\lbrace 1,2 \rbrace$ and
$\lbrace 7,8 \rbrace$ are in contact, and by $t=3$ (pink),
$\lbrace 1,2,3,4 \rbrace$ and
$\lbrace 5,6,7,8 \rbrace$ are touching.

Although the model is not fully specified, perhaps analogous processes have been studied in the literature? Essentially I want to know:

Q. What happens? Is there some limit configuration as $r$ grows?

Intuitively, it feels like all empty space would be squeezed out. Perhaps the behavior for points initially confined to a finite region $R$, as that region grows larger, would hint at the situation for the infinite plane? I'd appreciate pointers or observations—Thanks!

**Update**. Here is one of the many images from the
Lubachevsky, Graham, Stillinger work to which Henry Cohn points,
"Spontaneous Patterns in Disk Packings":