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. 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).
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.