Tech-chasing is the act of following or predicting an opponent's tech or floor recovery in order to attack them before they can respond. Because a character's tech animations have small windows of vulnerability before ending and allowing action, it is possible to read (or, in some cases, react to) the direction of a player's tech and punish them. Characters with fast movement speeds like Captain Falcon are very good at tech-chasing as they can reach the opponent's destination faster, though those with long-reaching attacks like Marth can do similarly well. Additionally, characters with high air speed can be good at tech chasing, especially on platforms, like Jigglypuff, and, in Melee, characters with a long Wavedash can be good at tech-chasing, like Luigi and Ice Climbers.
Since most of the high tier characters in Melee have a high movement speed, tech-chasing is very prevalent and important part of the punish game at a high level. In particular, Fox, Sheik, Captain Falcon, Marth, Peach and Luigi rely very heavily on tech-chasing in their punish games. Projectiles can also do a good job of tech-chasing when timed correctly. Tech-chasing is much easier to do on characters with short tech rolls (such as Olimar) than those with long tech rolls (such as Zelda). Additionally, whilst most characters, given a knockdown and the right position, can be tech-chased on reaction by characters that are fast enough, some cannot be. For example, Sheik has functionally identical tech-in-place and tech in animations until a point in them by which it is too late to react in time to the differences, so usually tech-chases against Sheik rely on a read or moves that cover both those options, like Raptor Boost.
Tech-chasing is possible on platforms and may be easier to perform on them depending on the scenario. Since the length of a tech roll is limited by the length of a platform, it is often easier to position to react to different options when tech-chasing on a platform. For example, in Melee, Sheik may use a throw that causes someone to tech on a platform, and can then either use aerials or quickly land on the platform to react to the next option; Jigglypuff notably can use her up-throw to set up a reaction tech-chase into Rest on platforms, making it especially lethal in her punish game.
However, platform tech-chasing does come with some limitations; several characters cannot cover every option on reaction at most percents. Fox is often reliant on using his up throw to put opponents into a knockdown on platforms and then hitting them with an up air to start large combos, but he can only do a 50/50 at lower percents, either reading a tech-in-place/missed-tech or waiting and reacting to either tech roll in or away. Additionally, in Melee, platform tech-chases that don't use throws are often vulnerable to Slideoff DI. A common example occurs with Marth, where until around 60% an opponent can utilising slideoff DI to escape his platform tech-chases using up-air or forward-air. Often these slideoff scenarios can lead to a big punish on the character attempting the tech-chase.
It is possible to tech-chase with a powerful slow-to-start move in certain situations, such as when the roll direction is predicted, or the defender is against an edge or a wall and ends up rolling towards it, therefore not moving. Teching in place (without rolling) can also allow this.
Some characters are able to incorporate tech-chasing into a pseudo-chain grab sequence. A classic example is Captain Falcon's up throw against fast-fallers in Melee. After he throws, Captain Falcon can gain another grab by successfully tech-chasing his enemy. A well-known example in Brawl is Snake's down throw. During the animation he lays his opponent on the ground and after the throw is over, if he can predict his opponent's roll, he can tech-chase them into another grab. In SSB4 and Ultimate, Ganondorf can use Flame Choke to tech-chase opponents, as the move is not affected by the game's grab immunity window.
Tech-chasing is less effective in SSB4 and Ultimate because the duration of tech rolls (alongside regular rolls in SSB4) has been reduced, giving less time for the tech-chasing player to react to their opponent's tech option. Additionally, chain-grabbing is impossible in these two games due to a mechanic granting a 70 frame (~1 second) period of grab immunity to opponents that have been grabbed or thrown.