Short Answer: Just falling to prone counts as movement and still requires to be on your turn, but it's something that can easily be setup as a reaction.
Dropping to prone is found inside the movement phase rules, which means that it is used in your turn's movement:
You can drop prone without using any of your speed.
So consider it an action that is in your movement action, but uses 0 ft. So even if you use your full 30 ft (or however much movement speed your character has), you can still drop to prone at the end, because it costs 0 ft of movement.
Now, I'm glad you brought up the Ready action, because that is exactly what you can do with it.
Sometimes you want to get the jump on a foe or wait for a particular
circumstance before you act. To do so, you can take the Ready action
on your turn, which lets you act using your reaction before the start
of your next turn.
First, you decide what perceivable circumstance will trigger your
reaction. Then, you choose the action you will take in response to
that trigger, or you choose to move up to your speed in response to
it. Examples include "If the cultist steps on the trapdoor, I'll pull
the lever that opens it," and "If the goblin steps next to me, I move
When the trigger occurs, you can either take your reaction right after
the trigger finishes or ignore the trigger. Remember that you can take
only one reaction per round.
This means that dropping to prone can either be done on your turn's move action, or during the move action of the reaction from readying an action outside your turn. To do this, you'll give your DM a trigger for your reaction to occur, then go prone. You can even use your movement speed first and then go prone, or go prone and crawl up to half your speed, because dropping to prone uses 0 ft of movement.
And yes, you can use your turn's move, and then move again on your reaction.