The Rooster

[BrianSoddingBoru4]

My understanding is evolution moves very slowly or not at all until that advantageous random variation hits. After that the specific adaptation usually spreads very quickly.

You're describing punctuated equilibrium (Eldredge and Gould) which caused no end of fuss when it was first presented.  Many of the stricter Darwinians had a lot of trouble accepting that the gradualism proposed by Darwin might not be the case (some still haven't).

Current thinking is that gradualism and PE are both true, depending on the circumstances in which groups of organisms find themselves.

Boru