Actually, this is not true. Neo-Darwinian theory says that species will be
adapted to their environments, but not "exquisitely" so. Evolution is the
ultimate pragmatist. Things are adapted to their environments "well enough."
Sometimes it *continues* to be well enough when the environment changes,
oftentimes not, and sometimes an adaptation turns out to be a major
advantage. But, adaptation only "cares" to go far enough to make further
adaptation nearly useless, and then drift sets in. Sometimes an adaptation
that starts out as just good enough turns out to be (perhaps after further
environmental changes) to be a *major* advantage (for example, the human
brain, which, for all I know, may have provided a relatively small advantage
to its first possessors).
In any case, "exquisitely adapted" is *far* too strong for the average case.
As to the climatic issue, I'll have to see the paper. however, I see no
reason to believe that evolution of climate-adaptedness would necessarily
occur *during* the climatic change rather than afterward, unless the
climatic change is very slow but ultimately fairly major. At first, the
natural capabilities of the animals might be enough. Major genetic
adaptation would set in in a noticeable way only when the climatic changes
were killing off a major portion of the population. And, obviously, if the
climatic change was *too* fast and too severe, it would simply kill the
species off in that area.