Essentially, dwarf leaders always commanded dwarves. They were never the leaders of any alliance of dwarves plus men and/or elves.
In the First Age, most of the great captains were elves, leading armies of elves alone or elves plus men or occasionally elves plus men and dwarves. Toward the end there were man captains as well.
In the Second Age, the man captains became more prominent and The Last Alliance of Men and Elves was pretty much jointly commanded by Gil-galad and Elendil.
In the Third Age, the elves started leaving Middle Earth for the Undying Lands in serious numbers and the race of man was ascendant. The great elf leaders - Cirdan, Elrond, Galadriel, and Thranduil - stay in place and defend their own territory. Same with the dwarves in Erebor and The Iron Hills. Only the mannish leaders take the war to Mordor.
That's the 30 second version of the lore reasons why there are no Dwarf (or Elf) Captains.
Cener, Ingo, Rilibald, Hesred, Halras, Loringo
Arkenstone (ex-Elendilmir) - The Osgiliath Guard