All Mystery Gifts inevitably expire when the servers shut down, even if they don't have an overt end date (e.g. the Pokedex rewards in Pokemon HOME will eventually no longer be accessible). There are ways around this for select events using the DNS Exploit, but those rely on fanmade servers and therefore will also vanish in the future.
The rest of this answer would depend on how broadly you define "events". Do in-game event Pokemon (e.g. starters) count? What about Pokemon Ranger's once-per-cartridge Manaphy?
