The idea of gradual evolution versus sudden change is relatively new. The recent (last 100 years) discoveries regarding plate tectonics, and giant space bombs has changed thinking. Species can go along for many thousands of years with very few useful mutations and thus few changes. But if conditions are suddenly/catastrophically changed, then rapid evolution can take place. Just looking at the evolution on the Galopagus Islands with different subspecies on each island shows that. Also, what may look very rapid when examining fossils might take many generations. For example, I have read that it takes about 10,000 years of deposition for 1 inch of sedimentary rock to form. If some birds breed after 2 years, then that would be 5,000 generations of bird fossils possible in 1 inch of stone. By the same calculation, if a human generation is 25 years, then it there would be 125,000 years of evolution represented by 5,000 generations. This puts us back in Neanderthal territory, with archaic sapiens in the mix. I believe the “punctuated” part refers to the gradual evolution having an exclamation point thrown at it with fast changing conditions and fast evolution to go with it.
The punctuation mark in punctuated equilibria is the diastrophe (no joke) -- it's a semantic ploy to avoid the term "catastrophe". Most of the time there's stasis, other than loss of half of the genetic material of each parent (in species that use sexual reproduction and are not hermaphroditic); mutations are evidently irrelevant, since long sequences of the code is identical from person to person.