Well, looking at the HMOVE chart, you can get *almost* there.
Well, maybe I can write a "horseshoes" or "hand grenade" game.
Although no emulators will support it yet, I think I have a solution for normal Ataris. With HMxx set to $80, do a very late HMOVE on the scan line I want to push right. Do another HMOVE a few cycles later, after the end of HBLANK. This should turn on the pulsers and leave them on. They won't have any effect, however, until the start of the next HBLANK. Wait for 16 pulses to be output, then hit another HMOVE to stop the pulsers. The sprites on this scan line will be pushed left 8 pixels.
Thus, three any-value stores for every two scan lines to shove all the sprites alternately left and right by 8 pixels. I doubt there's any cheaper way than that.
If it's necessary to have one of the sprites not move, that might still be possible, though I would expect four HMOVEs per scan line.
I would not expect these approaches to work at all usefully on the FB2, nor on the "funny" Juniors. On the funny Juniors, it should be possible to push sprites left 8 pixels in the usual manner (HMOVE at cycle 74 with an HMxx of 0). To push sprites right 8 pixels with an HMxx of 0, do a very late HMOVE with an HMxx of 0. To make a sprite not move, use an HMxx of $80.