In literature, Homer implies that Trojans and Greeks keep similar gods, albeit being different cultures. I also noticed on a map of Neo-Assyrian Ninevah there was a temple to Hermes. I'm afraid my ignorance prevents me from forming a well worded question, but I was hoping someone could discuss how and why a god might "spread" beyond its indigenous culture in the late Bronze Age.
I've only now had the time to give this question the time it deserves, so hopefully you're still interested in an answer.
#Pantheons
Each of the societies in the ancient Near East had its own pantheon of native deities. The Egyptians worshiped Amun, Re, Isis, Horus, Osiris, Thoth, Hathor, and so on. The most prominent deities among the Hittites were the Storm God of Hatti and the Sun Goddess of Arinna. Marduk reigned supreme in Babylonia; other prominent deities included Enki/Ea, Ninurta, Šamaš, Inanna, Enlil, Gula, and a host of others. We know much less about the Mycenaean pantheon, but the names of some gods known from classical Greek texts have been identified in Linear B texts, such as Poseidon (po-si-da-o), Zeus (di-we), Dionysus (di-wo-nu-so), and Ares (a-re). Additionally, there are deities whose roles are not quite clear, such as the “Lady of the Labyrinth” (da-pu2-ri-to-yo po-ti-ni-ya). The close ties between the people of a land and their gods are seen clearly in the Hittite prayer of King Arnuwanda and Queen Ašmunikkal to the Sun Goddess of Arinna.
Only Hatti is a true, pure land for you gods, and only in the land of Hatti do we repeatedly give you pure, great, fine sacrifices. Only in the land of Hatti do we establish respect for you gods.
#Movements of People and Their Gods
To complicate this picture, groups moved around and brought their gods with them. This happened with Semitic-speaking peoples, who moved into southern Mesopotamia, and the Hittites, who migrated into Anatolia. What to do with the deities? There were two common strategies for preventing religious conflict.
The first strategy was to combine deities into a single deity worshiped by multiple groups. The Semitic sun god Šamaš, for example, was identified with the Sumerian sun god Utu, and the two names are used interchangeably in Akkadian and Sumerian texts. As another example, in the Ptolemaic period, the Greek god Zeus was identified with the Egyptian god Amun. Scholars of ancient religion call this process “syncretization.”
The second strategy was to worship both the gods of the people(s) already living in the region and newly introduced gods. The Hittites are infamous for doing this. To the preexisting Hattic (or Hattian) panthon, the Hittites added their own gods derived from Proto-Indo-European religious traditions. Around 1400 BCE, when the Hittites exerted hegemony over Cilicia and northern Syria, the Hittites began worshiping Hurrian gods as well. To muddle things still further, the Hittites were inheritors of the Babylonian scribal tradition - in other words, Hittite scribes copied Babylonian hymns, prayers, and other religious texts - so Babylonian gods and even at least one Elamite deity made their way to the Hittite capital. Little wonder that the Hittites claimed to worship 1000 gods! The Hittite queen Puduhepa (ca. 1290-1220 BCE) attempted to overhaul and simplify this unwieldy pantheon, and her prayers indicate a process of syncretization near the end of the Hittite Empire. In a prayer to the Sun Goddess of Arinna, for example, she conflates the Hittite Sun Goddess and the Hurrian goddess Hebat.
O Sun Goddess of Arinna, my lady, queen of all lands! In Hatti you gave yourself the name Sun Goddess of Arinna, but in the land which you made, that of the cedar [i.e. Syria-Lebanon], there you gave yourself the name Hebat.
#Other Transmissions of Deities
I mentioned above that groups of people moved around the ancient Near East, bringing their pantheons with them. Sometimes, however, only a few deities were transmitted. As ancient empires developed, governors and soldiers were sent to monitor conquered regions and exact tribute. Additionally, prisoners-of-war were deported to the imperial core to work as servants/slaves. Consequently, ancient empires adopted new gods from abroad. Religious cults were also transmitted through more peaceful means, such as the movement of merchants and itinerant artisans and the transmission of literature. Finally, gods could arrive as part of the diplomatic marriages that were common in the Late Bronze Age. The Hittite king Muršili II, for example, mentions in passing that his stepmother Tawananna, the daughter of the king of Babylonia, carried out traditions from her homeland.
During the New Kingdom, Egypt established control over the southern Levant as well as close ties with other Canaanite centers like Ugarit. As a result of this contact, Canaanite deities such as Resheph, Anat, Astarte, and Ba’al were introduced to Egypt. These gods became such a prominent part of the Egyptian pantheon, in fact, that Ramesses II gave his daughter the Semitic name Bint-Anat, “daughter of Anat.” Moreover, Canaanite gods began appearing in Egyptian literature, such as the fragmentary tale about Astarte and the Sea (P. Amherst IX), which was probably adapted from Canaanite literature.
#Scribes and Gods
As a scribal convenience, the name of a foreign god could be written using local deities. The Egyptian-Hittite peace treaty provides the best example. Hittite treaties required both human witnesses (high-ranking members of the royal court) and divine witnesses, who enforced the terms of the treaty. As you’d expect, the Egyptian-Hittite treaty includes both Hittite and Egyptian deities. Interestingly, the Egyptians wrote the names of the Hittite deities using the names of Egyptian gods.
They are with me as witnesses to these words: the Re, lord of the sky, the Re of Arinna, Seth of the sky, Seth of Hatti, Seth of Arinna, Seth of Zippalanda…
This is a bit deceptive, as the Egyptians did not really think that Re and Seth were gods of the Hittites. The Egyptian sun god Re was rather similar to the Hittite Sun Goddess, however, and the Hittite Storm God could be equated with the Egyptian Seth. The Egyptians could therefore write “Seth of Hatti” (StX n xt) to refer to the deity the Hittites wrote as “the Storm God of Hatti” (dU uruHa-at-ti).
#Gods on a Mission
Gods were sometimes sent from one society to another on diplomatic missions. In one of the Amarna letters (EA 23), for example, the king of Mitanni notes that he sent the cult statue of the goddess Šauška on a mission to Egypt.
Thus spoke Šauška of Nineveh, mistress of all lands: “To the land of Egypt, to the country which I love, I very much would like to go, and then I would return.” Now I have sent her, and she has gone forth.
One sometimes gets the impression that local ritual practitioners did not know quite what to make of these foreign gods. A Hittite oracular inquiry, for example, inquires of the gods about the proper way to worship the cult statues from Ahhiyawa (Mycenaean Greece) and Lazpa (Lesbos) that had been sent to the aid of the ailing Hittite king Hattušili III.
With respect to the fact that the freeing of the deity of Ahhiyawa, the deity of Lazpa, and the personal deity of his Majesty was indicated by oracle as incumbent upon his Majesty—when they bring the personal deity of the King, should they bring them (the other deities) too? And as they perform the rite for them over the course of three days, is it likewise mandated for three days for the deity of Ahhiyawa and the deity of Lazpa? And as his Majesty has done obeisance to the polluted and purified offering tables and they have sacrificed in the style of Hattusa, should he do precisely the same for them?
For more on gods on diplomatic missions, see my write-up about the Bentresh stela and the Egyptian cult statue sent to aid the princess of Bactria in Are there any accounts from ancient literature which narrate demon-possessed victims being cured by an exorcist?
#Limits of Divine Power
Well-traveled or well-educated individuals knew that different societies worshiped different gods. In the prayer of Arnuwanda and Ašmunikkal mentioned above, for example, the king and queen note that only the Hittites worship the Hittite gods in the proper fashion.
Not everyone was willing to accept geographic limitations of their gods’ power, however. In the Egyptian Tale of Wenamun, the eponymous priest attempts to persuade the ruler of Byblos to give him cedar logs by emphasizing the importance of the god he brought with him.
There is no ship on the river that does not belong to Amun. His is the sea and his the Lebanon of which you say “It is mine.” …Truly it was Amun-Re, king of the gods, who said to Herihor, my master: “Send me!” And he made me come with this great god. But look, you have made this great god spend these 29 days moored in your harbor. Did you not know that he was here? Is he not who he was? You are prepared to haggle over Lebanon with Amun, its lord? As to your saying, “The former kings sent silver and gold” - if the kings had life and health, they would not have sent those things! It was in the place of life and health that they sent those things to your fathers! But Amun-Re, king of the gods, he is the lord of life and health, and he was the lord of your fathers! They passed their lifetimes offering to Amun. You too, you are the servant of Amun!
The prince of Byblos was left unmoved by this appeal to piety and refused to give Wenamun the cedar until payment had been received from Egypt.
A similar sentiment is found in the religious hymns of the Amarna period. Since Akhenaten largely rejected the existence of gods other than the Aten, hymns emphasize the Aten ruling benevolently over all peoples and lands. An excerpt from the Great Hymn to the Aten:
How many are your deeds,
Though hidden from sight,
O sole god beside whom there is none!
You made the earth as you wished, you alone,
All peoples, herds, and flocks,
All upon earth that walk on legs,
All upon high that fly on wings,
The lands of Khor (the Levant) and Kush,
The land of Egypt.