Where does it explicitly say that the Ban applies to races other than humans?
We already know that giant eagles (which are apparently mortal) travel back and forth. Of course, they are the Heralds of Manwe.
I don’t think Tolkien explicitlywrote that it applied to races other than Men. But it was Men who received the Gift of eventual death, and yet soon yearned for immortality. As Ar-Pharazon proved, it was Men who really needed to have the Ban in place. Dwarves don’t build seagoing ships, apparently, and (most) Elves know they’re going West eventually anyway.
*–then, at long last, the grey rain-curtain turned all to glass and silver, and was rolled back, revealing a far green country under a swift sunrise. The tiny boat, its sails hanging in tattered ribbons, drew ever nearer until at last it could go no further. Leaping from the prow and floundering through the shallow waters to shore, Isengar Took stood at last upon the diamond sands of the Blessed Realm.
‘I claim this new land for the Shire!’ he cried. ‘From this day on, it shall be known as New Tuckborough!’
His cry rose up even to the halls of Valimar; and all the Valar heard and were amazed, and even Manwe was flabbergasted.*
At last Ulmo spoke. ‘It would seem that we must now either sink all of Eriador into the sea, or else let us forget that this ever happened.’
‘What then shall we do with that-- er-- ?’ said Manwe, gesturing toward the shore, where the hobbit stood amid a lush grove of niphredil, relieving himself from his long journey.
‘Give him to Aule?’ Varda suggested hesitantly. ‘Doesn’t he enjoy such… short things?’
‘Oh please, can I have him?’ Yavanna cried. ‘I think he’s cute!’
‘Oi! You lot up on the hill!’ Isengar called out. ‘Anything to eat around here?’
‘Let us never speak of this again,’ Manwe decreed firmly.
flabbergasted = ME flappe, stroke or blow + agasten, to terrify (pp). Eggs for breakfast? Press pants.
–found scrawled on an empty packet of Walkers Potato Crisps behind the kitchen sink at 99 Holywell Street, Merton College, Oxford.
Terrifel, that was wonderful!
The ban of the Valar explicitly excluded the Numenoreans.
I suspect the Valar (who were remarkably short-sighted at times) would have just winged it if other peoples had started showing up.
As for other races, the dwarves and ents pretty much followed their core programming, and apparently Eru didn’t give the dwarves that much free will back when he adopted them.
Great post Terrifel. Pre-ban we know of Earendil and Tuor of the non-elven and post Numenor we know of Bilbo, Frodo, Sam and Gimli.
Earendil was of course Half Elven and chose Elvenkind. Tuor was a mortal but:
Jim
Yes, but somewhat against his will, or at least his inclination – didn’t The Silmarillion (which I haven’t got with me) say that when forced to choose, he identified more with his father’s people (Mankind) but chose Elvenkind to avoid being parted from his wife Elwing, who was similarly half-Elven but chose first?
I can see who wore the pants in that relationship! 
Earendil: Manwe, I choose… Mankind.
Elwing: No you don’t. It’s Elvenkind for us.
Earendil: Hey, I’m the one who built this boat and put my life on the line to sail here, I get to pick!
Elwing: But it’s MY Silmaril, and I want to be an Elf!
Earendil: And I want to be a Man!
Elwing: Oh, and leave me alone for the rest of Eternity? You’d better think again! Or the rest of your short, mortal life will be MISERABLE!
Manwe: Err… We’ll leave you two alone for a while.
(later)
Manwe: Well, Earendil… Are you a Man after all?
Earendil: (muttering)…No… I guess not…
I bow in your presence, that was truly awesome.