What does the slang moving right along mean?
If you were to say "moving along" it would mean making progress along some course of action. For example, "My painting is moving along," means that the painting I am working on is getting closer to completion.
If you change that to "moving right along" it means that you are making progress without any setbacks or interruptions. As Geoffrey Pullum describes it, you can use right as an adverb when there is a proper, perfect way to execute a verb. You can move right along, but you can't wander right around.
The way correctness links to the special preposition-modifying use of right, then, is that there can be an absolutely right way to instantiate a spatial or temporal relation (or metaphorical analog thereof). Right lays emphasis on the instantiation being exactly the right one for the job.