Moving can occur at the level of individuals or shifts in the whole population to a more favorable environment. This shift may also include changes to migration routes. But not all species can move to more favorable locations if those locations are not able to support the needs of the population. Take marine mammals for example, many require narrow temperature ranges and dissolved oxygen to survive.
Other ways animals adapt may include changes in behavior such as shifting times of day spent foraging, resting and mating. Animals may also increase the amount of time they spend cooling off.
Physiological adaptations to a changing environment can occur within individuals and populations as well as between generations. This can include shifts in proteins that are expressed, such as heat shock proteins which are responsible for protecting proteins from being misfolded when heated. If animals are not able to adapt or move, they may die.
Stillman JH. Heat waves, the new normal: Summertime temperature extremes will impact animals, ecosystems, and human communities. Physiology 34(2): 2019. https://doi.org/10.1152/physiol.00040.2018