One of the many storms-in-a-teacup of Evolutionary theory is the idea pf Punctuated Equilibrium, proposed by Gould and Eldredge. Rather than expound the details, here is
Wikipedia's reasonably clear take on the subject Dawkins has, as the Wikipedia article describes, attempted to belittle the theory as merely obvious becasue no evolutionary theorist would ever assert that evolution has to proceeed at a constant speed. To Dawkins, there is a continuous variation in the rate ate which species evolve, from so slow as to be effectively stopped to so fast as to appear to be a jump in the poorly-sampled fossil record. To Gould and Eldredge, there are two different clusters of speeds - slow, which fixes a smoothly changing fossil record, and fast ,which produces breaks in the fossil record which could be mistaken as saltationist.
I think the community at large has agreed that the truth is probably somewhere in between the two extremes. Furthermore, the disagreement has become embarassing because the Creationists (or Intelligent Designers or whatever flag they fly under these days) have taken to portraying a purely technical argument within a totally Darwinian academic comminity as some kind of deep schism which shows that Darwinians cannot even agree on their own theory. Since nobody wants to give the slightest aid to such falsehoods, the community has agreed that the benefits of discussing a minor flourish to Darwinian theory are outweighed by the succour given to the forces of irrationality, and it is better to keep quiet for a decade or two.
Nonetheless, I (in this case totally unembarassed by the slightest trace of repute in the field) am going to put my point of view - which is closer to that of Eldredge and Gould than that of Dawkins.
Eldredge and Gould built their view on the evidence of their speciality, paleontology. And in particular, on the admitted fact that it is very difficult to find transitional species. The fossil record tends to show many milennia of a species being essentially constant, and then it is suddenly replaced (or sometimes joined) by a related, but distinct, species. Their view is that this represents a sudden spurt in evolution. The view of their oppponents is eithr that it just represents the paucity of the fossil record, or it represents a spatial takeover by a species that eveolved separately, but relatively slowly, in a different area, and then invaded and took over the territiry in a geological blink of the eye. An example of the latter would be the displacement of the British Red Squirrel by the North American Grey Squirrel, which has take place over the last century or two. Paleontologically, this will appear instantaneous.
Military service is described as long stretches of boredom punctuated by moments of sheer terror. I view the life of a species (insofar as a such a diffuce concept can be said to have a life) as being much the same. For most of the time, ecosystems are in balance. All ecological niches are occupied, and each species is trying to maximise the yield of whatever niche it has. Of course, with the slow grinding of the years, one species may manage to outcompete, and steal the niche of, another; and for various reasons a species may split. Wet and dry hot and cold years may come and go, but the ecosystem is broadly in balance, and the species has fully exploited its niche.
But this does not mean that mutations have stopped. The DNA of the species is being updated at the same rate as always. Ofr course, as always, most mutations are negative and are dropped. And the ones that are not negative are probably, in a stable ecosystem, nearly neutral. Imagine one which makes the animal hairier. Well, it is probably at optimal hairiness for its current niche. So the appearance of a new, slightly hairier, gene, will probably simple depress the occurrence of the existing hairiness genes, so that the current phenotype remains approximately as hairy as before. And the same applies if a gene for less hairiness arises - the gene pool will be perturbed, but the phenotype (and hence its fossils) will be apparently unchanged.
And then something changes. Maybe a climate change; maybe a new territory opens up; maybe a new predator, a new competitor or a new food source arrives. Suddently, it is all change. The ecosystem itself is changing. Old niches are closing and, most importantly, new niches are opening. And niches are "winner takes all". If one species occupies an ecological niche,it is very hard for another to take it over. And which sopecies will find it easiest to take over the new niche? The one with the most genertic variation which it can use to optimise itself to that niche. Which means the one which has evolved least in recent millenia, and hence has the largest bank of unexploited variation to draw on. New species, which have recently emerged from small populations, will have relatively small amounts of variation to draw on, and will therefore be out-evolved by an old, apparently static, species that has a huge bank of diversity to draw on.
Hence, I would expect the typical pattern to be for a lineage to remain stable for a long time, and then to jump into a new niche, and hence become a new species. In doing so, they will "elbow aside" more recently developed species which do not have sucha a deep pool of exploitable variation. Hence, Punctuated Equilibrium.