Move On Dark won't actually be able to help you here. This is because Move On Dark only looks at what cues are coming next in this playback. Move On Dark won't be able to prepare fixtures ready for other playbacks that may be triggered.
So in this situation, I would manually record the start position for the moving lights in the cue before the "search light" playback is triggered.
To do this, go into the cue before the "search light" playback is triggered, select the moving light, and set it to its first position. Ideally if you recorded the first cue of your "search light" playback with a palette, you can apply this same palette, and your fixture will go to the correct position. Once in position, you can then update the cue. By default, "SmartTag" will be enabled in the console's Record Options. With SmartTag enabled, ZerOS will not include data for a fixture, if the fixture is @ 0%. So in this situation, you will need to disable SmartTag, so that the position information can be included with the light @ 0%. To do this, press and hold UPDATE to open the Update Options, and in here ensure that the SmartTag button has a blue stripe, meaning disabled. Then, tap your Master Playback's GO button to update the current cue.
Now upon going into the cue before your search light playback is triggered, the mover will prepare, and upon going into the next cue, the search light playback will be triggered with your mover ready in position.
Hope this helps.