The historical examples of near complete depopulation of a territory are extremely rare, probably Easter Island stands chiefly among them, for a combination of ecological and external factors:
It is believed that Easter Island's Polynesian inhabitants arrived on Easter Island sometime in the 12th century AD. They created a thriving and industrious culture, as evidenced by the island's numerous enormous stone moai and other artifacts. However, land clearing for cultivation and the introduction of the Polynesian rat led to gradual deforestation. By the time of European arrival in 1722, the island's population was estimated to be 2,000–3,000. European diseases, Peruvian slave raiding in the 1860s and emigration to other islands, e.g.Tahiti, further depleted the population, reducing it to a low of 111 native inhabitants in 1877. [...] The 2017 Chilean census registered 7,750 people on the island, of whom 3,512 (45%) considered themselves Rapa Nui.
Otherwise, there have been cycles of population boom and bust (of lesser magnitude) practically everywhere. (Not till long ago, the main concern was that population was growing too fast, and might still be the case world-wide.) The concept of population cycle is better understood in biology, for the obvious reason that the factors affecting non-human populations are simpler to study.
So it's pretty improbable that the downward trend is going to reach zero in some countries instead of a non-zero minima (and possibly rebounding). Except for the massive ecological disaster scenario, say global warming out of control, rendering some countries uninhabitable, everything else is too speculative in terms of complete depopulation, i.e. has no historical equivalent. But if we take Easter Island as example, a combination of fertility rebound and/or immigration seem to be the likely scenario.
What would happen politically is even more speculative. Easter Island was annexed, but it's not clear that that would happen to a larger country whose borders are guaranteed by international alliances etc.
A sufficiently scary scenario is many such countries becoming like Greece, i.e. long-term economic disaster zones, despite preserving their independence. However, that's not of immediate concern; despite topping the depopulation charts, most countries in Eastern Europe also top them in economic growth (at EU level), so it's not even clear when population crunch will start to affect their growth.