Something that Bluejay said when writing about rats made me think about evolutionary stasis in omnivorous animals capable of surviving on a variety of foodstuffs, that are very behaviourally adaptable already. In such a species, wouldn't it be a stable, unchanging environment that actually selected for change?
Imagine a population living in an environment that changes rapidly from generation to generation - such as those humans create, perhaps. If things are very changeable, there'll be little sustained selective pressure in any particular direction. Bigger animals may be favoured in one particular generation, and then smaller ones again a generation or two down the line as conditions change. We'd see slight variations around a point, but little major change.
Too much specialisation in any one direction would be penalised by selection, as a specialisation which is useful for a couple of generations may turn out to be deeply counterproductive shortly afterwards. The most highly adaptable animals - those who will readily alter their behaviour in new conditions and can move to new food sources easily, will consistently pass on their genes generation after generation. The changing environment is selecting for the adaptable individuals, maintaining stasis in an already adaptable species.
If we imagine that one population got isolated somewhere remote, however, where the environment is very stable over long periods of time - it would be this that prompted change in the population. There would be no selective pressure to retain the ability to metabolise unavailable sources of food, and if more efficient metabolism of whatever is available is possible at the expense of being able to digest almost anything, specialisation will be selected for. Equally, a mind and body designed for behavioural adaptability might not be as successful as one more narrowly focused on exploiting the resources of the new environment, assuming these to be constant, and specialisation would again be selected for.
If this idle speculation has any merit then, it's a changeable and uncertain environment that would select for relative stasis in the most adaptable organisms; while stabilising that environment would prompt evolutionary change (at least until the organism's as specialised as it's going to get for while).