General:
If he tends to wait at the ledge and dash at you before attempting the grab, feel free to dash/wavedash back and grab him or hit him with a quick, strong attack (i.e. Luigi dsmash or Puff fsmash), or do a retreating aerial, it's hard to punish those out of shield if he blocks then grabs or whatever, and it's a bit harder for him to crouch-cancel punish. If he shoots lasers until you approach, and you're playing a character with a reflecting move, just reflect until he stops, otherwise, try to powershield, or if he shoots high lasers, go under them (most dashes and dash attacks/grabs do a good job of this but most dash attacks are unsafe at low%, proceed with caution).
lol, up until this point, I just was imagining and assuming this was on FD. If it isn't you can just wait on platforms, or potentially use projectiles of your own. If he's staying close enough to the edge that he's just bthrow -> dairing you and not dashing back before he does the grab, you shouldn't have to worry about him dashdancing much farther back, so you don't have to focus on overshooting your approach as much. This means if you're doing an aerial approach, you can jump earlier, and have a more secure, lower, safer aerial. Since you'll be waiting longer, if he tries to roll before you attack back to center stage, you have a bigger window to waveland back and punish his roll, or just generally be at the advantage outside of stage positioning in the neutral that follows.
Back to FD image (though most of this would still apply on the other stages):
If he's doing perfect pivots/wavedashes or whatever to the ledge as fast as possible, stalling and perfect ledge dashes back on like if Mew2King was a Falco main you might want to try baiting some kind of movement back on stage with a dash dance and punishing, otherwise you probably shouldn't challenge it.
Character Specific.
Since you're apparently playing Puff, know that you can challenge aerial space with auto-cancel nairs (high and low) if he shields and tries to hit you if you just float above him or something. DIing off stage as low% as Puff shouldn't get you killed by Falco, so just do that and if you go low, get back up without going into a range he can safely drop below the ledge or short hop off, hit you and recover (this isn't that big a range). Puff has one of the better dash attacks for just getting people off stage, if he's unable to shield at a give point, this move is fast and you can respond with it unless he's capable of dairing, or you'd be running into a Smash attack or something. Additionally, if he tried to crouch cancel, there's a decent chance he'll fall to his demise with one of Falco's long aerials or air dodge. You can do a bair wall and be sure to space bairs and retreat with them, most people have a hard time dealing with them. Puff has great aerial mobility and Falco's dash isn't very fast, so you can just float almost into his range to try to bait something out and punish with a grab or a lot of other things depending on what he throws out. You can also just float offstage and he'll have limited ability to hit you with anything but lasers.
For Luigi, lasers are the bane of your existence. Stay in a range you can threaten him so he can't use them. Punish if he does anything. If he's being passive, mix up between spaced retreating fair (dair and bair also work), low nair, grab, and a down-b where you hit him or his shield then go center stage. The low nair should be safe, starts a combo if he doesn't block, lets you escape and punish oos options if he does, grab is the counter to block and you can gimp Falco or start a chaingrab/tech-chase, if you do the down-b, hit him early (because you started close) and immediately retreat to center stage if he blocked, he has limited ability to follow and get a strong punish, and he'd have to give up his gimping position. If he tried to follow you after blocking to punish, you can come back with the down-b to hit him from a wd oos or jump, and you can start a combo or tech-chase off of this. You can also do this if he didn't block the first hit. All of these options have different timings that he'd have to try to hit or grab you and would make it hard for him to just sit there and grab. If you get grabbed, try to DI away to avoid the dair, and save your jump. You could try to go high above him toward center stage, but there' a good chance he'd just bair you and reach the ground before you did if you air dodged.
If he's content to sit there and never laser, just sit center stage, or powershield until one hits him, then camp the other side of the stage, or on a platform if available. Or, take him to the once legal Mute City, where he can't ledgecamp no more.
