If it seems the dynasties moved around a lot, that is in no small part because China has a very long history, and the factors favoring a particular location at one point may not be the same five hundred years late. The moves were always for a specific reason--you won't get a general answer because none exists, and not every dynasty did move the capital. The Tang Dynasty located its capital at Chang'an, the most famous capital of the Han. But, for a quick rundown:

The Tang was at Chang'an because of the historical prestige of the site, but things had changed in the millenium since Qin Shi Huang had located his capital there (in no small part because it had historical associations with the Zhou). For one, the city was much larger, and needed to import enormous quantities of food from far off, and in many ways it was in an unfavorable location for this as it required transport through the Hangu Pass--not impossible, but a headache, and eventually this outweighed the advantages associated with its defensibly. Even during the Tang, the center was shifting towards Luoyang, an with the collapse of the Tang imperial structure the city of Chang'an was not sustainable (Mark Edward Lewis' Tang Dynasty goes into this in detail).

Also during the Tang the economic center started to shift towards the south, and the imperial capital depended ever more on rice shipped up from the Yangtze. This rice would travel up through the canals and on its way to Chang'an (through Luoyang) it would pass through Kaifeng (Bianzhou on that map). This is a very favorable location, because it has the political advantages of a northern location, but easier access to southern grain and rice. The chaos following the Tang was the ruin of Chang'an, and so when the Song assumed power Kaifeng was the obvious choice (it's more complicated than this). However, in 1127 the Mongols conquered North China, making Kaifeng no longer a sustainable capital city. During the Northern Song, Hangzhou, which you can see was at the bottom of the canal, had flourished in no small part due to overseas trade, and so with the fall of Kaifeng was an obvious place to move the capital to.

Beijing is actually a terrible location for a capital city (it is strategically located, but even harder to get needed food supplies to from the south than Chang'an), I am actually not certain why the Mongols remained there after the conquest of the south, or why the Manchu stayed there. The Ming is a bit more interesting: the first emperor located the capital at Nanjing, located at a very favorable location in the lower Yangtze (a bit up river from where the Canal crosses) but the third emperor (the Yongle Emperor) after deposing his brother moved the capital back to Beijing because he, correctly, saw that Nanjing was unfavorable territory. The reputation has grown that Nanjing was the city of the scholar-official (shi dafu) while Beijing was the city of the eunuch bureaucrat--this is probably unfair but gives you a sense of what people think.

These topics will be discussed in most major surveys on Chinese history--John King Fairbank's China: A New History is always a good place to start.

EDIT: Geography

I can't speak on behalf of all dynasties and their capitals but most often in the case of Beijing it is because the emperor or dictator at the time often has base of power in the north.

The Mongol Yuan dynasty was the first to make Beijing their capital, which makes sense as they first invaded the northern Jurchen Jin dynasty, taking their capital before invading the southern Song. When the Ming overthrew the Mongols they reverted back to the traditional capital of Nanjing only to switch back to Beijing. So why the change?

The first Ming emperor moved the capital south, closer to its Chinese roots and gave his son Beijing, the former capital. When his son took the throne as emperor Yongle (after a bloody coup) he reverted it back to Beijing. His justification is that it is closer to the northern front, making it easier to respond to future "barbarian invasions." While this reasoning is true, it is also likely his intentions were that he wanted to move the capital to his power base (since he had been holding Beijing on behalf of his father for up till his ascension).

When the Ming fell to the Qing, the capital of Beijing once again fell to northern invaders. The Manchu of the Qing made Beijing their capital while subduing southern China. Once again the capital was closer to their source of power (in this case Manchuria).

The Qing fell to the republic and the new democratic revolutionaries moved the capital again to Nanjing, the more historic and central capital. However soon after Sun Yat-sen was made president he was forced to give up his presidency to the most powerful General in China, Yuan Shikai. Yuan Shikai was the general of the North China army (Beiyang army), so naturally one of his acts as presidents was move the capital to the northern capital, and away from the power base of his political opponents, the KMT.

Eventually the KMT under Chiang Kai-shek would take power, moving the capital back to southern Nanjing (also worth noting that the KMT military wing had its origins in the south as well). As the focus of my study is pre-modern China I can only suspect in the case of the communists; I am open for correction. Though I suspect that when the communists fought the KMT they once again followed the pattern of a power base in the north. The Communists had a strong presence in north China since the long march and in Manchuria since the Japanese withdrawal. Also, the northern capital would put them closer to the USSR, nominally and ally and tactically a threat.

TL;DR: capitals move depending on the origin of the dynasty or emperor to where he has most power

I'll focus on two occasions that represent two very different reasons for moving the capital: the Song Dynasty xingzai capital of Hangzhou (Lin'an) and the move of the Jin Dynasty capital south to Beijing (Zhongdu).

The xingzai, temporary capital, of Hangzhou represents one of the most common reasons for moving the capital: War. There were a succession of temporary residences that the Song retreated to while escaping from North China, which had been overtaken by the Jin. The Jin, founded by the Jurchen tribes of Manchuria (and ancestors of the Manchus), fought a decades long war with the Song. They captured Kaifeng (Bianjing), the capital of the Song dynasty, held the Chinese emperor captive, and sent the emperor and other members of the royal family to Manchuria as prisoners. One of the few to escape was Gaozong (Zhao Gou), a sibling of the captive emperor. He moved from city to city, from Yingtian to Yangzhou to Hangzhou. Hangzhou was designated the the temporary capital of the Song instead of an official capital, a symbolic gesture to show that the Song had every intention of retaking North China from the Jin. They never did.

The Jin move of the capital to Beijing represents another important reason for moving: Political legitimacy. The emperor responsible for moving the capital was Hailingwang. Hailingwang worked towards controlling all of China, and changing the capital to one south of Manchuria was one way of showing that he meant it. The old capital of the Jin was demoted to a prefecture and Hailingwang had the Jurchen palaces in the city razed. A few years later, he invaded the Song. He never did become emperor of the whole of China, he was not a popular leader and his assassins put a stop to that (heck, he even had his own stepmother executed for criticizing his policies), but Beijing was capital of the Jin from then on. At least, up until the Mongols besieged Beijing and the Jin moved to Kaifeng.

Sources:

Jin dynasty chapter of The Cambridge History of China: Volume 6, Alien Regimes and Border States (1994) by Herbert Franke

Song dynasty chapter of The Cambridge History of China: Volume 5, The Sung Dynasty and Its Precursors (2009) by Jing-shen Tao

The Jurchen in Twelfth-Century China (1976) by Jing-shen Tao

Imperial China: 900–1800 (1999) by Frederick Mote.

EDIT: Formatting.

The capital cities were seen as reflections of imperial power, and so the city itself would be seen as arising out of the emperor's will, which conversely meant its fall would reflect the demise of imperial authority.

You see this pattern repeated over and over with each new regime either pingponging back and forth between existing capitals, or recreating brand new capital cities either from scratch, or near the old site of previous capitals.

To give you an idea of this pattern, you have Qin Shihuang founding his dynasty capital at Xianyang. It was sacked during the Qin dynasty's decline and the capital changed to Changan across the river, founded by Han Dynasty founder Gaozu. After the replacement of Western Han with Wang Mang and his subsequent fall, Changan was sacked, and the capital moved to Luoyang. After Eastern Han's fall, Luoyang was sacked by Dong Zhuo and the capital moved back to Changan. After the Cao Wei dynasty tookover, they moved the capital back to Luoyang. Once the Sui came to power, they moved the capital BACK to Changan.

This of course doesn't even get into all the movements and creations of empires in the later imperial dynasties like Kaifeng and Beijing.

As reiterated, the movement of the capital was a political as well as practical act. It was seen as a physical representation of the new regime, and also because of the geographic nature of Chinese politics given how often new dynasties forced existing aristocrats to move with them to new capitals away from the aristocrats old power bases, it was also the political imposition of imperial power.

And a more simpler explanation, of course, was that this was made easier by the fact that the Chinese prefer their buildings to be made out of wood. Which made reconstruction significantly quicker, especially since in the movement of these capitals, as previously mentioned, the whole administrative apparatus would be moved, from the aristocrats, to bureaucrats, to the merchants, to the workers.