How to transport shot glasses

Boxes with dividers!

You could buy a couple pieces of Styrofoam, fit them into a box, and cut holes just big enough to hold each one! Something similar to this. Easy to put them in the box and remove them, though possibly time consuming to initially setup the Styrofoam.

Or wrap in bubble wrap (cleaner than newspaper)

Oooh, I had a similar issue when moving a little less than 2 months ago! We stacked the shot glasses and then wrapped the stack in bubble wrap and laid it down sideways in the box. Did that as many times as needed to get all the shot glasses in there and filled the rest of the box with some kitchen towels. You could also use more bubble wrap or newspaper to fill the rest of the box.