interveni

Both pieces are yours already. Inter is between. Veni is to come. Interveni is to come between, which is what intervening is, and the English word says the same thing in Latin where you cannot see it. This is the best kind of Esperanto word: an international form that a learner could have built unaided, so learning it costs nothing and confirms the machine at the same time. Compare interrompi, to break between, which is to interrupt, and intermeti, to put between, and you have a small productive pattern rather than three separate memory items.

A preposition is not a prefix, and Esperanto does not care. Inter is an ordinary preposition, not one of the official affixes, but Rule 11 lets any two words be joined into a compound, so a preposition can lead one whenever the sense works. That is how kunveni (to come together, to meet), elveni (to come out), alveni (to arrive) and deveni (to originate) are made, all on the same verb. Getting veni and its family into your head is worth more than most vocabulary of the same size, because the prepositions do the rest. The full mechanism is in compounding.

How it behaves. Interveni is intransitive: you intervene en a situation or inter two parties, and it never takes a direct object. Interveno is the intervention, intervenanto the party doing it, interveni favore al iu to intervene on somebody's side. The register is neutral and reaches from the playground to the Security Council, exactly as in English, and its quieter cousin enmiksiĝi, to mix oneself in, is the one that means meddling.