For example Alexandrou Limen, Alexandreia Eschate or Alexandreia Oxou. Did they just disappear with time? What happend with the native greeks? What was the cultural interaction between the populations? Why did Alexandria in Egypt become such a successful city but the others dont even exist anymore?
Around 140 BC, the city died. To judge from the arrowheads scattered around the great walls, the final struggle was fierce. Once the invaders breached the defenses, however, the end was probably quick. The temples and their colossal statues were burned, as were the palaces and the gymnasium. Most of the inhabitants were enslaved or scattered. And within a few years, the Greek city we call Ai Khanoum, in the green fields by the rushing Oxus in what is now northern Afghanistan, had vanished.
Ai Khanoum may have been Alexandria Oxiana, one of the many settlements scattered across the Greek far east by Alexander and his successors. Plutarch claimed that Alexander founded no fewer than 70 cities, many in central Asia and India (the actual number is probably less impressive). With a bit of poetic license (unsurprising, since he never went anywhere near the area), Strabo called Bactria (roughly northern Afghanistan and the adjoining bits of Uzbekistan, Tajikstan, and Pakistan) a land of 1000 cities. Some of these cities, like Bactra (modern Balkh, Afghainstan) and Maracanda (Samakand, Uzbekistan) were great and prosperous. Most were relatively modest. Virtually all, with the exception of Ai Khanoum, are unexcavated.
Before we discuss the little we do know about the Greek cities of central Asia, it might be useful to survey the political history of the region. After Alexander's death, his eastern conquests were incorporated into the kingdom of Seleucus, who reigned over an empire stretching from Aegean to the Hindu Kush. Seleucus gave the easternmost reaches of his domains to the Indian empire-builder Chandragupta Maurya in exchange for 500 war elephants (which turned out to be a good deal, since Seleucus used those elephants to win the critical battle of Ipsos). Seleucus preserved his central Asia domains, however, and continued to encourage Greek settlers - both retired soldiers and adventurers from old Greece - to settle in the fertile lands of Bactria. But under Seleucus' successors, the governors of Bactria became increasingly independent; and from about 250 BC onward, they ruled as independent kings of a large and prosperous realm.
The Bactrian kings gradually pushed the boundaries of their realm north into central Asia and south into India, where an independent Indo-Greek kingdom was established. The culmination of Greek power in central Asia came in the reign of the mighty Eucratides, whose domains ranged from the steppes of Uzbekistan to the jungles of the Punjab. The rise of Parthia, however, severely weakened the Bactrian kingdom; and starting around 145 BC, a series of nomadic incursions delivered the coup de grace. The Indo-Greek kingdom in the south survived a century longer, and some of the Greek cities in Bactria apparently continued to flourish. As late as the first century CE, the Kushan Empire (created by descendants of the nomadic peoples who had conquered the Greco-Bactrian kingdom) still used Greek as an administrative language, and the famous Greek-inflected art of Gandhara suggests that some Greek cultural influences lingered even longer.
To review: there were a substantial number of cities with Greek names, a hegemonic Greek culture, and at least some ethnically Greek inhabitants were scattered from central Asia to India. For almost two centuries, these cities existed in a Hellenistic political matrix; after this dissolved, the cities remained, Greek islands in a "barbarian" sea. So what happened to them?
In most cases, we simply don't know. The fact that the Kushans kept using Greek suggests that quite a few Greek speakers remained in these settlements for centuries after the dissolution of the Greco-Bactrian kingdom. As far as we can tell from the very scattered archaeological record, the nomadic incursions of the second century BC were not enormously disruptive. Most settlements probably just opened their gates to the invaders, paid them tribute, and continued to exist much as they had. The Greek inhabitants had always been a small minority; and to judge from the remains at Ai Khanoum, they lived alongside non-Greek populations that may have learned the Greek language, but retained their traditional cultures. After the disappearance of the Greco-Bactrian kingdom, these ethnic Greeks and their culture were slowly absorbed by their neighbors, leaving widely-scattered traces in such unexpected contexts as Buddhist sculpture (substantially influenced by Greek models) and Chinese tapestries.
We know little about the specific fates of Greek cities, since few sites have been systematically excavated. The most famous material traces of the Bactrian Greeks are the fabulous coins found in every corner of central Asia. There have been limited excavations in Balkh, Samarkand, and a scattering of other sites. The only Greco-Bactrian city to be systematically explored, however, is Ai Khanoum. You can read a full description of the city here, so I'll just describe the highlights.
Ai Khanoum was a substantial fortified settlement, apparently established to protect the Bactrian heartland from nomad incursions. Despite some interesting buildings that blended Greek and central Asian traditions, the city plan and primary monuments were Greek in inspiration, down to a theater and a gymnasium inscribed with maxims of the Delphic Oracle. The city's destruction, described in the thrilling first paragraph of this overlong post, has traditionally been ascribed to the nomads who ended the Greco-Bactrian kingdom. Recent scholarship, however, has suggested that the city was destroyed in two stages: first an uprising, in which local populations joined the nomads in attacking the city, and the royal palace was destroyed; and later a more wholesale destruction, apparently at the hands and hooves of the nomads, that left the entire city in ruins.
Was Ai Khanoum typical? We have no way of knowing. But we do know that it, and the other cities of Bactria, were cultural anomalies once the Greek kingdoms vanished. As a result - and unlike the bustling harbor metropolis of Alexandria in Egypt - they were cut off from the cultural and economic centers of the Greek world, and eventually adopted the languages and mores of their neighbors. This happened quite early in the Indo-Greek kingdom, where widespread conversion to Buddhism is attested. There may have sometimes been hostility been the Greeks and their neighbors, as at Ai Khanoum. The typical fate of the Greek cities, however, was probably a slow and unspectacular fade into the cultural background.
The settlements left behind from the conquests of Alexander eventually came to form the Greco-Bactrian Kingdom, one of the more forgotten hellenistic kingdoms to have arisen from Alexander's conquests.
After his conquests and death, the empire Alexander had formed was split into four kingdoms, each with varying lifespans. The land between Anatolia, present day Turkey, and modern day India became the Seleucid Empire which lasted in some form or another until the 1st century BC, however parts of the empire seceded or were conquered by other forces. This is where the Greco-Bactrian Kingdom arose from, by 256 BC the local governor of Bactria, Diodotus, seceded from the Seleucid Empire.
The Kingdom itself lasted for several hundred years and, in my opinion, gave rise to some of the most interesting mergers of art and styles during the ancient period. Because of the proximity and large amounts of trade and an overlapping population with the ruling parties in, at that time, was the Mauryan Empire in India, the ruling Greeks began to adopt a number of hindu and buddhist art forms and styles. This is evidenced in the coinage of the era which, both in the Mauryan empire and the Greco-Bactrian kingdom, Hellenistic and Hindu iconography began to merge.
This artistic influence remained not just in the realm of political iconography and numismatics, there came a large influx of so called Greco-Buddhist art. At this point in time, some Greeks in the Greco-Bactrian kingdoms began to adopt Buddhism with lead to the development of Buddhist art which had hellenistic styled influences. There are still remnants of this influence today in some depictions of the Buddha. (There is still some debate on how influential the hellenistic styles were in potentially information changes in buddhist theology at this time.) One of the later Indo-Greek/Greco-Bactrian kings, Menander, is known to have been, at the minimum, a patron of Buddhism and Buddhist art, being referenced in several buddhist theological texts. The coins of Menander's reign are extremely prevalent and show the combination of Indian and Greek language and culture/religion.
The reason for this slow assimilation and blending of styles came from the manner in which the Greeks retained power in the conquered territories, preferring to allow those local governors, satraps, who allied with Alexander to remain in power and not destroy the power system which existed in the area. Greek was paired with local languages for governmental business and trade remained mostly uninterrupted through the territory, barring the external issues with trade as the Seleucids amoung other powers faced external pressures.
The Greco-Bactrian Kingdom began to decline and eventually disintegrated likely through dynastic turmoil and civil wars as well as pressures from local tribes/rulers and the other Indo-Greek Kingdoms. The remaining Indo-Greek Kingdoms lasted until around the close of the 1st century BC with the remaining political structures dissolved and the hellenistic kings as a political entity no longer existed due to invasions from other powers in central asia.
Others can speak more to the Greco-Bactrians, but specifically around some of the Alexandrias in Central Asia, it's worth noting that many of these settlements have continued on in some fashion down to the present day. The ruins of Alexandria Eschate are in the middle of of the city of Khujand, Tajikistan; the ruins of Alexansria Arachosia are in Kandahar, Afghanistan, Alexandria in the Caucasus eventually became Bagram, and another Alexandria became Merv.
So why are these towns and cities less central in size and importance than Alexandria, Egypt? It mostly has to do with strategic location and trade routes. Many of these towns have been very important administrative centers for various kingdoms and empires after Alexander, and Merv in particular is thought to have been one of the biggest cities in the world, by population, before the Mongol conquests. That city went into a long decline, before being abandoned some time in the 18th century, with the modern city of Mary founded nearby in the 19th.
These cities had much of their wealth and status tied to the overland Eurasian trade routes that are known to us as the Silk Road. As sea routes began to increase in traffic and prominence in the 16th century, these land routes became less central, and a lot of these cities' fame and fortune dried up. Khujand is known for its region's cotton crop today, instead of being an entrepot for international trade, which it once was.
Alexandria, Egypt long kept that central role, especially as the port for Egypt on the Mediterranean, and the important geographic place Egypt holds in between Mediterranean sea traffic and sea routes on the Red Sea and in the Indian Ocean.
In summary: the Greek influence in many of these Central Asian Alexandrias has long faded, and relatively speaking these cities have declined in importance, but many if not most of them are still there, have been mostly continuously inhabited since Alexander's time, and still have local importance.