Geological unrest could be considered any period in which there is dramatic change. This includes such long-term shifts as tectonic plate movements, mountains building and crumbling, and short-term and often dramatic geological upheavals such as earthquakes, volcanic eruptions and climate change.
Evolution is a mechanism for adaptation to change. With no change, there can be no evolution. As geological changes occur, habitats change and those living things that fail to adapt to the change die off, and those that adapt or have mutations that allow individuals to survive the change evolve and adapt. So evolution is a process that can not only endure during geological 'unrest', but flourishes during times of change, and even needs geological change to provide a stimulus for evolution.
Some geological forms of 'unrest' are short-term, such as volcanic eruptions and earthquakes caused by such things as sudden slippages of tectonic plates. Evolution cannot occur short term. Faced with a flow of lava, creatures either die or escape, but there is no time for evolutionary adaptation.
Geological changes occurring over a longer period do allow for evolutionary adaptation, however. For example, the earth's climate is constantly changing, and our current climate is much cooler than it has been for long periods in the past. There are also much bigger seasonal differences now than earlier. In the Late Cretaceous (about 100 million years ago) for example, the climate was cooling, and the fossil records show that the flora and fauna began to change as flowering plants in Alaska (still joined at the time to what is now Russia) began to replace the sub-tropical plants such as ferns, cycads and gingkos that had flourished in the warm period. Some species adapted to the cooling by evolving seeds that could survive the winters, whereas others died out. Meanwhile, in what is now Antarctica, the vast forests began to die out there too and be replaced by other flora as the south also began to be covered in ice.
Another example is the evolutionary changes accompanying the development of the Isthmus of Panama, which grew from an enlarging chain of islands joining North and South America, in the late Pliocene. The new land link led to mammals in both continents moving between the two. The armadillo, for example, moved into North America, where it still survives in places. A successful migrant in the opposite direction was the sigmodontine rodent group that evolved into 45 different genera in South America. As always, as new niches become available, life takes every opportunity to adapt to those niches.
Many geologic changes occur over almost unimaginably long periods of time, but this is also the time scale of evolution. Tectonic plate movements, building and erosion of mountains, and so on occur so slowly that they are hard to view as geological 'unrest', but it is clear that evolution 'endures' through these changes, as is demonstrated by the similarities in flora and fauna in different continents that were once joined.
In the Triassic and Early Jurassic periods, there was a single landmass, Pangea, which split about 130 million years ago into two supercontinents: Laurasia (which became North America, Europe and Asia), and Gondwana (which became South America, Africa, India, Antarctica and Australia). Several families of marsupial are known to have crossed to Antarctica and thence to Australia while they were both joined as Gondwana. No placental mammals are known to have made the crossing. The unique fauna of Australia developed from the few marsupials that crossed, while their relatives in South America became extinct. In Australia, the marsupials had no predators and therefore survived and adapted to every ecological niche, evolving into dozens of marsupials such as kangaroos, wallabies and euros. Everywhere else, early egg-laying mammals died out, but in Australia, they remain.
There is no question that evolution 'endures' during geological unrest. Adapting to change and unrest is the actual driver of evolution.