postkuri

It says what it does. Post is after and kuri at rank 393 is to run, so postkuri is to run after. English has the same construction sitting in plain sight and still reaches for the French-derived chase half the time; Esperanto simply glues the preposition on the front and gets on with it.

The grammar changes when you glue it. Kuri is intransitive: you cannot run something. Put post- on the front and the compound becomes transitive, so the thing you are chasing takes the accusative: la hundo postkuris la katon, not al la kato. This is one of the tidiest examples of a prefix changing verb government, and the transitivity article works through the rest of the family.

The fence with sekvi. Sekvi at rank 150 is to follow, and it is calm: you can follow somebody down a corridor at walking pace, or follow an argument. Postkuri has running in it, and with running comes urgency and usually the risk of not catching up. Kapti is the successful end of it. Ĉasi is hunting, which is chasing with intent to kill.

And a story where nobody has to. In La vulpo the fox takes the butter without breaking into a trot: Ĝi eĉ ne devis kuri, it did not even have to run. That is the joke of the whole tale, and it is worth reading once you can hear the difference between running and running after.