It depends on the context
Also probably your DM's opinion of the word "perceivable"
A number of other people have quoted the language of the readied action ability, and called specific attention to fact that the trigger for a readied action must be a "perceivable circumstance", and that the earliest your readied action can happen is after the trigger. (The Player's Handbook, pg. 193)
That definitely seems to rule out,
"I ready my action, casting booming blade before they take their move action" because how is your character supposed to perceive a move action?
But if you stated it a little bit differently, as "I ready my action, casting booming blade before they run out of my range," things start to get murkier. If you unpack things out to, "I ready my action, casting booming blade when they start to move, before they run out of my range," it gets hard to see why anyone would take issue. The one thing that can get weird is that this kind of trigger can involve the sorts of determinations that usually your DM would call for a skill check or save for (e.g. "Roll dex to see if you can grab your friend before they fall into the inky depths"), but I'm pretty sure that is a product of the rules themselves, not evidence of incorrect play.
The issue here is that the trigger needs to be a "perceivable circumstance". But there are no rules clearly outlining what is or is not a perceivable circumstance in the game of D&D 5e. There are rules for special cases -- your DM may ask you to roll Perception in certain vaguely defined contexts, and there are rules for how you can't hear things when you are in the area of effect of a Silence spell, for example. But there is no mechanical test for, "is x a perceivable circumstance?"
Even worse, there does not seem to be a RAW interpretation for how perceivable circumstances fit into the turn structure of combat, so it is unclear if, for example, your readied action trigger can interrupt (read: happen within/alongside/before) another character's action in the way that an opportunity attack "interrupts the provoking creature’s movement, occurring right before the creature leaves your reach." (Id., pg. 195)
The "perceivable circumstance" language really only tells us to match the triggers to the fiction, not the mechanics. The reason that trying to trigger off of a move action seems pretty obviously wrong based on that language is because we implicitly assume that things like "move actions" and "ability checks" don't really exist in the in-game universe.