In Tolkien's universe, what happens if, for example, an elven Ring of Power is worn by a dwarf, or vice-versa? We, seemingly, only have mention of this happening to One Ring, but I also don't remember any mention of it being made for a specific race.
There are five classes of rings mentioned: The One Ring, the Three held by the Elves, the Seven held by the Dwarves, the Nine that went to Mortal Men and others "lesser essays in the craft."
'In Eregion long ago many Elven-rings were made, magic rings as you call them, and they were, of course, of various kinds: some more potent and some less. The lesser rings were only essays in the craft before it was full-grown, and to the Elven-smiths they were but trifles - yet still to my mind dangerous for mortals. But the Great Rings, the Rings of Power, they were perilous.
many eyes were turned to Elrond in fear and wonder as he told of the Elven-smiths of Eregion and their friendship with Moria, and their eagerness for knowledge, by which Sauron ensnared them. For in that time he was not yet evil to behold, and they received his aid and grew mighty in craft, whereas he learned all their secrets, and betrayed them, and forged secretly in the Mountain of Fire the One Ring to be their master. But Celebrimbor was aware of him, and hid the Three which he had made;
The One was made by Sauron for Sauron, but its power could be used by whoever bore it if that person had sufficient will power.
Did not Gandalf tell you that the rings give power according to the measure of each possessor? Before you could use that power you would need to become far stronger, and to train your will to the domination of others.
It was more durable than the Seven or the Nine since four of the Seven were destroyed by dragon fire,
Seven the Dwarf-kings possessed, but three he has recovered, and the others the dragons have consumed.
nor was there ever any dragon, not even Ancalagon the Black, who could have harmed the One Ring
The Three were made by the Elves for their own use "they were forged by Celebrimbor alone" and had special properties "those who had them in their keeping could ward off the decays of time and postpone the weariness of the world". They were never worn by anyone but Elves and Gandalf, and we have no hard information what a Man or Dwarf could do with them.
'The Three, fairest of all, the Elf-lords hid from him, and his hand never touched them or sullied them.
they were not made as weapons of war or conquest: that is not their power. Those who made them did not desire strength or domination or hoarded wealth, but understanding, making, and healing, to preserve all things unstained.
This suggests that they were also different from the Seven and the Nine.
The other rings were made by the smiths of Eregion led by Celebrimbor under Sauron's tutelage and were captured by Sauron and given by him to Men and Dwarves. They each had their own decoration,
"The Nine, the Seven, and the Three," he said, "had each their proper gem. Not so the One. It was round and unadorned, as it were one of the lesser rings; but its maker set marks upon it that the skilled, maybe, could still see and read."
Other than the somewhat ambiguous Ring Inscription, there is no suggestion in Tolkien's writings that the Seven or the Nine were made for specific creatures, nor is there a suggestion that they were intrinsically different in their effect. It appears that the Rings of Power other than the One and the Three were undifferentiated:
But Sauron gathered into his hands all the remaining Rings of Power; and he dealt them out to the other peoples of Middle-earth, hoping thus to bring under his sway all those that desired secret power beyond the measure of their kind. Seven Rings he gave to the Dwarves; but to Men he gave nine, for Men proved in this matter as in others the readiest to his will.
The different effects of the Seven and the Nine on Dwarves and Men seems to have come from the nature of Dwarves and Men:
The Dwarves indeed proved tough and hard to tame; they ill endure the domination of others, and the thoughts of their hearts are hard to fathom, nor can they be turned to shadows. They used their rings only for the getting of wealth; but wrath and an over-mastering greed of gold were kindled in their hearts, of which evil enough after came to the profit of Sauron.
Men proved easier to ensnare. Those who used the Nine Rings became mighty in their day, kings, sorcerers, and warriors of old. They obtained glory and great wealth, yet it turned to their undoing. They had, as it seemed, unending life, yet life became unendurable to them... And one by one, sooner or later, according to their native strength and to the good or evil of their wills in the beginning, they fell under the thraldom of the ring that they bore and under the domination of the One, which was Sauron's.
One caveat to this. The Dwarves believed that the ring given to Durin was special:
Of this Ring something may be said here. It was believed by the Dwarves of Durin's Folk to be the first of the Seven that was forged; and they say that it was given to the King of Khazad-dûm, Durin III, by the Elven-smiths themselves and not by Sauron, though doubtless his evil power was on it, since he had aided in the forging of all the Seven.
We know nothing about the lesser rings other than that Gandalf still considered them to be dangerous, but less so than any of the rings of power.
Gandalf seems to suggest that any mortal could wield any of the Rings of Power:
'A mortal, Frodo, who keeps one of the Great Rings, does not die, but he does not grow or obtain more life, he merely continues, until at last every minute is a weariness. And if he often uses the Ring to make himself invisible, he fades: he becomes in the end invisible permanently, and walks in the twilight under the eye of the dark power that rules the Rings. Yes, sooner or later - later, if he is strong or well-meaning to begin with, but neither strength nor good purpose will last - sooner or later the dark power will devour him.'
- The One was made by Sauron alone and is unique.
- The Three were made by Celebrimbor alone and are very probably unique.
- The Seven and the Nine were made by the Elves with Sauron's assistance and as far as we can tell they were sixteen Rings of Power which were alike except for their gems.
- The "lesser rungs" were, well, lesser and certainly different from the One, the Three or the Sixteen.
Bottom line: There are hints that anyone could wield any ring, and there is no clear evidence that any of the rings of power couldn't be wielded by whoever possessed it. But the effect of a ring seems to have depended strongly on the nature of the wielder.