Keep in mind that you are talking about a very long period of time, with many different Canaanite kingdoms. The migration @MrDibble’s link above details happened until about 3,500 BC, at which point you could probably say that Canaanite history had begun. By the Iron Age, around 1200 BC, Israelites had emerged as distinctive enough cultures that you could think of them as something else. But the Phonecians continued referring to themselves as Canaanites going into the Iron Age - “Phonecian” is an exonym.
*If we think about this cladistically - birds evolved from dinosaurs, so birds are dinosaurs, but when someone uses the term dinosaur they often mean non-avian dinosaur, right? Perhaps Israelites can each be thought of as a group akin to “birds”, who are part of the larger group of Canaanites while also being identifiably distinct.
So, with that out of the way, let’s talk about your question.
There was both cooperation and warfare, as we might suspect when we are looking at such a long period of time.
For the most part, in the early Bronze Age (which basically starts in this region at the same time that the Canaanites do), there were no big unified Canaanite polities. Each Canaanite polity was its own city-state. This meant that Canaanites were vulnerable to domination by their larger and more urbanized neighbors, the Egyptians and the Hittites. And indeed, most Bronze Age history saw foreign rulers controlling the region.
There were exceptions, for example in the latter Bronze Age a dynasty of Canaanite kings managed to establish themselves on the throne of Egypt. But they basically became Egyptian pharaohs with a different ethnic background, they didn’t rule Egypt from Canaan or anything like that.
The individual Canaanite kingdoms would often use these neighbors to compete with one another; for example, a vassal king of the Hittites might switch sides to join the Egyptians in exchange for Egypt’s support in gaining concessions against a neighboring Canaanite kingdom.
Eventually, some of these kingdoms did get bigger, and often they competed, subjugating each other (often with foreign aid) and then losing control of their neighbors (often to rebellions with foreign aid).
The Late Brinze Age Collapse, in about 1200 BCE, gave the Canaanite kingdoms the chance to establish themselves independently of Egyptian or Hittite influence.
To the north, in what is today Lebanon, coastal city-states grew more powerful, rich, and influential as they mastered the Mediterranean trade networks that returned in the Iron Age following the Bronze Age Collapse. Eventually Tyre dominated the rest of the Phonecian city-states; from there, the Phonecians started colonizing Spain, North Africa, and the Mediterranean islands. The most important of these colonies would be a city they founded on the North African coast which they called simply “The New City”, or Qart Hadasht. (Hadash, incidentally, is the modern Hebrew word for “new” to this day!). Qart Hadasht would be Latinized as Carthago, and after Phonecia proper was conquered by Babylonians and later Persians and Greeks, Carthage would take control of the other Phonecian colonies in the western Mediterranean and become the largest empire in the region… for a time. Unfortunately for them, they ended up getting tangled with another rising regional power - the Roman Empire.
By this point, Carthage’s faith had become syncretic, bringing in Greek deities they’d been exposed to through trade. But the Romans were extremely offended by the Canaanite aspects of the Cartheginian faith, leading to the famous stories of child sacrifice to Baal Hammon. Historians are pretty divided over the reality of these stories - there is some archeological evidence for human and child sacrifice in Carthage, but that interpretation of the evidence has been challenged by some. I think the consensus right now is that some child sacrifice was practiced, but not to the extent that Roman sources describe.
Regardless, perhaps because of this attitude by the Romans, almost nothing of Punic culture survives. We have a couple of translated works (including one about an explorer who allegedly sailed past the Pillars of Hercules (Gibraltar) and down the African coast, describing the sun setting strangely in a way that matches what he would have seen had he crossed the equator and meeting a tribe of hairy men he dubs Gorilla!) and IIRC one Carthaginian poem that was included in a Roman(?) play about a Punic character.
So Carthage is one branch of the Canaanite tree, and we have followed it to where it ends. (Sort of. While the story goes that Rome burned Carthage to the ground and salted the earth around it, this did not actually happen. Rome eventually rebuilt Carthage and it became an important city, and Punic culture didn’t disappear overnight; it blended with Roman culture, continued blending with local Libyan culture, and therefore continued to influence the humab story. But it disappeared as a distinct and independent cultural tradition.)
The Phonecian homeland meanwhile went through its own trials and tribulations. First they were conquered by the Assyrians, and then the Babylonians. During this time Phonecian cities often revolted and were sieged down again. But eventually the Persians came in and, as elsewhere in their empire, acted as sorts of liberators. They set up some puppet kingdoms which were given remarkable autonomy; trade and prosperity resumed.
During this entire time, Phonecians still think of themselves as Canaanites. Their own name for themselves is Canaanites.
Pressure for this to change began when Alexander the Great conquered the region, due to the policy of Hellenization. However Hellenization was not pushed as heavily in Phonecia as it was in other parts of the Empire, at least under Alexander. The region became a central part of the Seleucid successor state, with Antioch often serving as the capital.
Finally, Phonecia would be conquered by Rome, and this would be the final nail in the coffin for a culturally independent non-Israelite Canaanite group. Romanization finished what Hellenization started, and the Canaanite faith faded into the Roman pantheon, and Christianity would pf course obliterate any remaining trace of such paganism.
That’s just the tip of the iceberg, of course. We have a long way to go to reach modern Lebanese or Lybian people, and we haven’t even touched on the emergence of the Israelites (first mention by Egypt appears to be right around the Late Bronze Age Collapse) and Judeans, their rising influence in central and southern Canaan, the way that Assyrian and Babylonian conquests drove their adoption of monotheism, the way that monotheism drove the rise of Jewish people as a separate ethnic identity than Canaanite, and what eventually happened to them.
But in the interest of writing a forum post and not a book, I will stop there.
Well, OK, no, I will share one more quick story.
As I hinted a couple of times, monotheism really took off during the Assyrian and Babylonian occupations, likely as a response to them.
But not all Jews were conquered by the Assyrians. Some fled the Assyrian Empire’s forays into Canaan, ending up in Elephantine, a city on the Nile in upper Egypt. And in fact, while they worshipped the same God that modern Jews do as their chief god, they also honored other Canaanite and Egyptian deities. They even built a temple to this God pn an island in the middle of the Nile, where they offered animal sacrifice (in more modern Judaism, this should only occur at the temple in Jerusalem, of course).
This community survived in Elephantine long after Judaism became monotheistic in Israel itself, which is a fascinating situation to think about, especially since the Achaemenid Empire would bring both Israel and Elephantine under Persian rule, and apparently at least some communication actually occurred during this period.