Regarding the carrying of the tabernacle, vessels, etc. the "carrying" also evolved "care." However, the Levites performed many other roles.
The question is complicated by the fact that although the Levites were supposedly scattered among the 12 tribes, they are also shown as performing important new duties in Jerusalem during the time of David. See Joshua 13:33: "To the tribe of Levi, Moses had given no inheritance; the Lord, the God of Israel, is their inheritance, as he promised them." Also note Gen. 49:7: "Cursed be their anger, for it is fierce, and their wrath, for it is cruel! I will divide them [Levi and Simeon] in Jacob and scatter them in Israel." But 1 Chronicles 15 shows them, under King David, as carrying the ark, acting as singers, gatekeepers, etc. According to critical scholars, this may be anachronistic, as there is no such description in Kings, and Chronicles reflects a post-exilic viewpoint when Levites played a more significant role in Jerusalem.
Under later kings: 2 Chronicles 23 portrays the priest Jehoiada as using armed Levites to execute a coup against the Baal-tolerant Queen Athaliah.
You priests and Levites... 6 Let no one enter the house of the Lord except the priests and ministering Levites; they may enter, for they are holy, but all the people shall keep the charge of the Lord. 7 The Levites shall surround the [young soon-to-be] king, each with his weapons in his hand; and whoever enters the house shall be slain. Be with the king when he comes in, and when he goes out. 8 The Levites and all Judah did according to all that Jehoi′ada the priest commanded... 13 Athali′ah rent her clothes, and cried, “Treason! Treason!” 14 Then Jehoi′ada the priest brought out the captains who were set over the army, saying to them, “Bring her out between the ranks; any one who follows her is to be slain with the sword.”
King Hezekiah used Levites to help cleanse the Temple of Jerusalem of allegedly idolatrous furnishings and carry them away (2 Chron. 29):
16The priests went into the inner part of the house of the Lord to cleanse it, and they brought out all the unclean things that they found in the temple of the Lord into the court of the house of the Lord, and the Levites took them and carried them out to the Wadi Kidron... 18 Then they went inside to King Hezekiah and said, “We have cleansed all the house of the Lord, the altar of burnt offering and all its utensils, and the table for the rows of bread and all its utensils. 19 All the utensils that King Ahaz discarded during his reign when he was faithless, we have restored and sanctified; see, they are in front of the altar of the Lord.”
2 Chron. 29 also mentions Levites as singers and musicians in the Temple, on the foundation of David's using them as such in the Tabernacle:
25He [Hezekiah] stationed the Levites in the house of the Lord with cymbals, harps, and lyres, according to the commandment of David and of Gad the king’s seer and of the prophet Nathan, for the commandment was from the Lord through his prophets. 26 The Levites stood with the instruments of David and the priests with the trumpets. The whole assembly worshiped, the singers sang, and the trumpeters sounded; all this continued until the burnt offering was finished. 29 When the offering was finished, the king and all who were present with him bowed down and worshiped. 30 King Hezekiah and the officials commanded the Levites to sing praises to the Lord with the words of David and of the seer Asaph.
2 Chron. 29 also introduces the Levitical function of assisting the priests in preparing sacrificial offerings. "But the priests were too few and could not skin all the burnt offerings, so, until other priests had sanctified themselves, their kindred, the Levites, helped them until the work was finished, for the Levites were more conscientious than the priests in sanctifying themselves."
For the period after the exile, Levitical functions are described in 1 Chron. 9. They include gatekeepers, care, and inventory of sacred utensils, supervising the kitchens including baking, and singing. They were also, in a certain sense, treasurers:
24The gatekeepers were on the four sides: east, west, north, and south; 25 and their kindred who were in their villages were obliged to come in every seven days, in turn, to be with them, 26 for the four chief gatekeepers, who were Levites, were in charge of the chambers and the treasures of the house of God.
Ezekiel's description [Eze 44] of the Levitical role adds the slaying of the sacrificial offerings, but it also denigrates the Levites for idolatry in attending the high places of Israel previously.
11 They shall be ministers in my sanctuary, having oversight at the gates of the temple and serving in the temple; they shall slaughter the burnt offering and the sacrifice for the people, and they shall attend on them and serve them. 12 Because they ministered to them before their idols and made the house of Israel stumble into iniquity, therefore I have sworn concerning them, says the Lord God, that they shall bear their punishment. 13 They shall not come near to me, to serve me as priest, nor come near any of my sacred offerings, the things that are most sacred, but they shall bear their shame and the consequences of the abominations that they have committed. 14 Yet I will appoint them to keep charge of the temple, to do all its chores, all that is to be done in it.
To summarize: the biblical account describes the Levites during the Temple periods as continuing their prior functions as porters and caretakers of the Temple furnishings. However, it adds other functions, including purifiers, bodyguards, gatekeepers, singers, musicians, butchers of sacrificial offerings, bakers, treasurers, and attendants to the priests.