Some answers of this question claim that after the world was changed it was forbidden by Eru for living beings like Elves and Men to sail from Aman to Middle-earth. The Istari, being Ainur, were permitted to do so.
Added March 16, 2022:
Some answers here also claim that return form Aman was forbidden after the changing of the world:
Tolkien discussed whether Glorfindel returned to Middle-earth during the 2nd Age or during the 3rd Age with the Istari. I don't know if he made up his mind.
How did the Istari come to Middle-earth during the Third Age? Did they fly
there in spirit form and then take human-like bodies, or did they take human like bodies in Aman and then sail to Middle-earth in vessels such as Elves or Men would have used?
In The Two Towers, book Three, Chapter IV "Treebeard":
"saruman is a wizard," answered Treebeard. "More than that I cannot say. I do not know the history of Wizards. They appeared first after the Great Shps came over the Sea; but if they came with the ships I can never can tell..."
Since 1966 I have assumed that those "Great Ships" came to Middle-earth about Third Age 1000, and the wizards came aboard them. Unless the wizards used magic to sail the Great Ships along the Straight Path from Aman to the world and over the oceans of the world to Middle-earth, there should have been sailors aboard those ships, and many sailors if they were "great" ships.
Those sailors would have been ainur using temporary phsyical bodies, or Elves, or a mixed group. And if there were any rebodied Elves in Aman who wanted to return to Middle-earth, and if they could get permission, they might have traveled as other passengers to Middle-earth on the Great Ships.
But today, March 16, 2022, when thinking about this passage, I wondered whether the Great Ships could have appeared at the shores of Middle-earth many centuries or even millennia before the wizards were first noticed by Treebeard.
Maybe those ships were Numenorean ships coming to Middle-earth during the later two millennia of the Second Age, or maybe they were the nine ships mentioned in an ancient rhyme of lore about the faithful Numenoreans who fled from the Downfall and landed in Middle-earth about Second age 3319.
Tall ships and tall kings
Three times three,
What brought they from the foundered land
Over the flowing sea?
Seven stars and seven stones
And one white tree.[note 1]
To someone as old, and as little interested in the outside world as Treebeard, a difference of a few thousand years might not be enough for him to think there couldn't be any connection between the two events.
Gandalf apparently first appeared at the Grey Havens, supporting the idea that the wizards sailed in ships from Aman to Middle-earth.
The Return of the King, Appendix B, The Tale of Years, mentions the coming of the wizards at the beginning of the section on the Third Age.
...The ring of Gil-Galad was gven by him to Elrond; but Cirdan surrendered his to Mithrandir. For Cirdan saw farther and deeper than any other in Middle-earth, and he welcomed Mithrandir at the Grey Havens, knowing wence he came and wither he would return.