After the Babylonian exile, the vast majority of the ten northern tribes didn’t even bother to come home. They had completely abandoned their identity as part of the nation of Israel and thus as God’s chosen people. They had become like the nations around them in diet, language, and pagan religion and just stayed in Babylon. Most of the nation of Judah however, returned to the land in the south to rebuild Jerusalem and the temple and to begin temple worship again. And that, I think, is the background for John the Baptist eating locust and wild honey.
It was the Judahites who were exiled to Babylon.

The northern kingdom was attacked by the Assyrians, although the
Neo-Assyrian Empire had indeed been conquered by the
Neo-Babylonian Empire, and Neo-Babylonians had in turn been conquered by the Medo-Persians (the
Achaemenid Empire).
According to 2 Chronicles 15:9, members of the tribes of Ephraim, Manasseh and Simeon fled to Judah during the reign of
Asa (913/10 BC-873/69 BC). Whether these groups were absorbed into the population or remained distinct groups, or returned to their tribal lands is not indicated.
In the thirty-ninth year of Azariah king of Judah, Menahem the son of Gadi began to reign over Israel, and he reigned ten years in Samaria. …] Pul the king of Assyria came against the land, and Menahem gave Pul a thousand talents of silver, that he might help him to confirm his hold on the royal power. Menahem exacted the money from Israel, that is, from all the wealthy men, fifty shekels of silver from every man, to give to the king of Assyria. So the king of Assyria turned back and did not stay there in the land. …] And Menahem slept with his fathers, and Pekahiah his son reigned in his place.
In the fiftieth year of Azariah king of Judah, Pekahiah the son of Menahem began to reign over Israel in Samaria, and he reigned two years. …] And Pekah the son of Remaliah, his captain, conspired against him with fifty men of the people of Gilead, and struck him down in Samaria, in the citadel of the king’s house with Argob and Arieh; he put him to death and reigned in his place.
In the fifty-second year of Azariah king of Judah, Pekah the son of Remaliah began to reign over Israel in Samaria, and he reigned twenty years. …] In the days of Pekah king of Israel, Tiglath-pileser king of Assyria came and captured Ijon, Abel-beth-maacah, Janoah, Kedesh, Hazor, Gilead, and Galilee, all the land of Naphtali, and he carried the people captive to Assyria. Then Hoshea the son of Elah made a conspiracy against Pekah the son of Remaliah and struck him down and put him to death and reigned in his place, in the twentieth year of Jotham the son of Uzziah. …]
In the twelfth year of Ahaz king of Judah, Hoshea the son of Elah began to reign in Samaria over Israel, and he reigned nine years. And he did what was evil in the sight of the LORD, yet not as the kings of Israel who were before him. Against him came up Shalmaneser king of Assyria. And Hoshea became his vassal and paid him tribute. But the king of Assyria found treachery in Hoshea, for he had sent messengers to So, king of Egypt, and offered no tribute to the king of Assyria, as he had done year by year. Therefore the king of Assyria shut him up and bound him in prison. Then the king of Assyria invaded all the land and came to Samaria, and for three years he besieged it.
In the ninth year of Hoshea, the king of Assyria captured Samaria, and he carried the Israelites away to Assyria and placed them in Halah, and on the Habor, the river of Gozan, and in the cities of the Medes.
With the Assyrian invasion of 722-20 BC and the deportation of Israelites came the end of the northern kingdom. It would seem though that not all were deported: 2 Chronicles 30:1-11 mentions northern Israelites (in particular, members of Dan, Ephraim, Manasseh, Asher and Zebulun) during
Hezekiah’s reign and how members of the latter three went to worship at the Temple in Jerusalem at that time. Israel Finkelstein recently estimated that only a fifth of the population (about 40,000) were actually resettled out of the area during the two deportation periods under Tiglath-Pileser III and his successor Sargon II.