I think one could make a case, if I understand the question correctly, that this is what happened to the Roman Empire. After the western part of the Empire, centered in Rome, where the empire originally started, fell into decay as a result of Germanic tribes invading, the East, Centered in Constantinople, flourished for quite some time.

The terms "country" or "nation" can have particular definitions in political science, but allowing for broad or general meanings -

The Roman Empire could be classed under this. Its capital was moved from Rome to Constantinople in 330 and as an institution it definitively ended only when the Ottoman Turks conquered Constantinople in 1453, centuries after Rome or, indeed, every other part of Italy had long since been lost by the empire to various successor states.

Migratory peoples are other possible examples. Those that come to mind are the Germanic tribes who moved into the Roman empire and carved out kingdoms out quite sizeable areas of it. The Visigoths took over the Iberian peninsula from the Romans. The Vandals eventually crossed the Mediterranean and conquered Roman North Africa.

Over in the New World, the migratory ancestors of the Aztec empire brought their wanderings from the north to an end when they established themselves in the Valley of Mexico.

Then there was the Umayyad Caliphate, which began with its capital in Damascus in the 7th century, which eventually had its final kingdom centered on Cordoba in far away Al-Andalus (Muslim Spain) in the 11th.

Again, depending on how one defines those terms, one might argue that China-Taiwan is another example. After the nationalist government under Chiang Kai Shek had been defeated and driven out of the mainland by Mao's communists in 1949, the former set itself up in Taiwan, insisting that it remained the same entity as the true and legitimate government of China while in practical fact exercising authority only in Taiwan, but not in China.

The Hungarians (Magyars) would be one of the more obvious examples of this. As they were moving into the Carpathian Basin (which is modern day Hungary and Romania) their older territories in what is now Moldova and Ukraine were being attacked by the Bulgarians and Pechenegs, causing the Magyars to abandon them and move into Hungary proper.

The Republic of China, not to be confused with The People's Republic of China. In 1949 Chiang Kai Shek fled to Taiwan with his government. They still posture like they are the legitimate Chinese government.

You could say this about migrant societies. The Roma left their homeland hundreds of years ago and still maintain a cohesive culture.

I'd suggest the Hurons. Their original range was along the Great Lakes, where they got exterminated by the Iroquois through proxy warfare for the French and the British. The surviving Huron-Wendat canoed down the St-Laurence river to their french allies in Québec City, where they were given land north of QC (Wendake). They have lived and thrived there to this day.

The Visigothic Kingdom was first established in 418, with Roman permision, in Aquitaine (SE France) and had its capital in Toulouse. As the Romans began to lose control of the Iberian Peninsula, they invited the Goths to move in and secure it for them. Then the Romans fell, leaving the Goths in control of both Aquitaine and most of Iberia.

Later in 507, at the battle of Vouillé, the Franks defeated and completely routed out the Goths, also killing their King, Alaric II. This disaster resulted in the Goths losing Aquitaine to the Franks, including Toulouse. Gothic power moved south and after a few decades Toledo, Spain was chosen as their new capital.

The Sovereign Military Order of Malta is a legal sovereignty (they issue their own passports and stamps, and retain observer status at the UN). While sovereign, they do not hold any land.

The order began in Jerusalem, where they ran a hospital and were known as the Knights Hospitaller. After being ejected from Jerusalem, they were given control of the island of Rhodes. After being ejected from Rhodes, they were given the island of Malta. Napoleon came along and ejected them from Malta, leaving the order with no land whatsoever. Currently, they maintain only an embassy in Vatican City.

There are also the Jews, who lost Israel for almost two millennia before returning.

Can the Mongol empire be classified into this?

Normans. They originated in Scandinavia and ended up controlling Normandy and Sicily. Then they controlled England after they lost both of those territories. Then they just melded into the native French, Italian, and Anglo-Saxon populations where they had previously ruled.