The Song arguably did this, but with a bunch of asterisks at the end of that sentence. They were pushed out of their northern lands by the Jurchen invasion and fled south eventually setting up their capital in what's now Hangzhou. I say arguably because they actually did control the southern area that they moved to, so it's not like they were re-establishing themselves on foreign soil. Still, it did involve a wholesale move of the government and everything had to be set up anew. It was a big enough change that they're often historically referred to as two different dynastic periods, so I think it may count.

As far as being better off, the circumstances were such that they pretty much were falling apart until their final demise at the hands of the Yuan. There were advantages to having the southern location, but as for whether they would have been in an objectively better situation had it not been for the Mongols, I can't really say. They never really got over the loss of their northern territories and attempts to reclaim the land cost them.

The Republic of China, since you brought it up, has a couple interesting points worth mentioning. Before the move to Taiwan at the end of the war, the Nationalists actually moved their capital in land from Nanjing as the Japanese advanced. It was pretty short lived, but part of the deal was that a lot of the major universities also moved and set up shop in land.

The same thing happened with the universities when the Nationalists relocated to Taipei, and now we have many universities here in Taiwan that have counterparts in China. After the PRC was more established they re-opened the campuses that the universities had abandoned a decade or so before. Despite the political tension between the two countries today, many of these schools cooperate with their sister school across the straights with both teacher exchanges and student exchanges.

Again as far as being better off, this again is not without interference from other factors. Taiwan is clearly better off as far as standard of living and stability than they were in the 1930s, but then they aren't fighting a war like they were then. Things were always pretty rocky for them on the mainland right from their start in 1911. To answer the "better off" question gets into too many historical what-ifs.

The Ancient Phoenicians never had a unified empire, in their homeland of modern Lebanon. Instead, it was a collection of small city states that shared a language and culture. Tyre, Sidon and Byblos were the three most powerful Phoenician city states, but they were constantly under threat by larger and more powerful empires to their east. The Assyrians, Babylonians and Persians all had a varying amount of control over the Phoenicians at various points of time, in the first millinium BC.
In 814 BC, Tyre established a colony in modern Tunisia called Carthage. Since the natives in Tunisia were not as powerful or as well organized as the Assyrians, Babylonians or Persians, Carthage was able to establish a far more powerful state than Tyre was ever able to do. Carthage was the dominant power in the Western Mediterranean until the three Punic Wars with Rome, led to the destruction of Carthage in 146 BC. Here is a map showing the extent of the Carthaginian empire in 264 BC.
http://en.wikipedia.org/wiki/Ancient_Carthage#mediaviewer/File:CarthageMap.png

Not exactly a whole empire but one of the most famous warlords of Japan, Tokugawa Ieyasu, did something similar. Essentially, the most powerful warlord of Japan, Toyotomi Hideyoshi, offered Ieyasu eight provinces in the Kanto region of Japan, or the area around modern day Tokyo, in exchange for the five provinces Ieyasu had control of. One of those provinces was his home province of Mikawa.

Now, the time we're talking about is the Sengoku period in Japanese history (1470ish-1570ish); a period of nearly constant warfare. Moving your power base away from your ancestral home was risky, incredibly so. However, the move worked out brilliantly for Ieyasu. Eventually, after Hideyoshi's death, Ieyasu came to found the Tokugawa Shogunate, which ruled over Japan for 260 years.