All species may be caught up in an evolutionary mega-system of life, but some are much less affected than others. The coelacanth did not change very much (except to get larger) in 60 million years. That is probably a record for minimal change in known vertebrates, but the phenomenon of minimal change over millions of years is more common with small invertebrates.
The record-holder for multicelled creatures may be Neopilina, a weird little mollusk with a circular internal body plan, measuring about 35mm long, which was thought extinct until dredged up on the famous Galathea expedition of 1952. The existing species Neopilina galatheae is almost unchanged from fossils that go back over 400MY - and were believed to have been extinct 350MY ago.
A newer example (see title link) is this fig wasp from the Isle of Wight - almost unchanged after 34 million years.
Why do some animals find a comfortable niche and stay there, while others in the same niche, or similarly "comfortable" positions in nature, get displaced and move to the "extinct" column? It's a complicated question to which the answer is not yet clear. One of science's countless mysteries.