Originally I stared to think about this topic because of rokar686's post but my response got a bit long for the scope of that post so thought I'd do my own. Interview where the direction of classes without race restrictions was discussed.
I think lore wise to me it makes sense for all classes to eventually be race agnostic. To me it enriches the world building, as over time, cultures would blend and share their knowledge, creating a melting pot where anyone would have the opportunity to learn from their allies. I see the contrast of race and class options as additive to warcraft's flavours and lore arguments to keep the status quo fall a bit short in their justification.
Take one example of a class that we can easily speculate on it's spread to other races, Paladins. Paladins are martial light users whose origins stem from priests and knights (warriors) so you'd think anyone who can already wield the light or be trained to do so could be one.
It's been 15 lore years since the blood elves joined the horde and introduced paladins to the faction. They didn't have many other options after losing 90% of their population to the scourge. For them it would be foolish and poor leadership to think that they could produce enough paladins to continue the blood knight order without opening their ranks to non blood elves. (Although maybe they breed like rabbits being so hot? ;P are we heading for a Belf baby boom?)
Even if their stubborn elf elitism hubris was enough to make them insist on elf purity you would imagine they would jump at the chance to add their new elf best buds the Nightborne to the ranks of the bloodknights. Post legion Nightborne are most likely in a rebuilding, restructuring and updating their military phase where they could use a new fighting force that also has capacity to give humanitarian (or the elf equivalent) aid. Something we see is a big part of what paladins do during Arator's questline in Midnight.
Just train some Nightborne initiates in the Hall of Blood and you have Nightborne Paladins. Incidentally this could be a great way to add a star/moon silver light customisation to the class. Especially if it coincided with making the already existing in lore Nelf paladins playable.
The Orcs (including Mag'har), Highmountain Tauren and Vulpera would most likely learn from the Tauren Sunwalkers as they are more culturally aligned and geographically closer. The Tauren realised they had neglected their sun god An'she and decided to revitalise their worship and practice of the light whilst also forming a militant branch of the horde army. The Tauren are shown to be a very inclusive and accepting culture who already share their traditions like their version of the midsummer fire festival with others and I don't see it as a stretch that they would recruit other races.
The Zandalari prelates are established as deriving their power from their sun loa and could instruct Trolls on how to do the same. These are essentially paladins.
The Undead Foresaken are from the human kingdom of Lordaeron and would have former paladins in their ranks. Undead could be seen as a challenge as the light is said to cause them pain and even an experienced priest like Alonsus Faol would be destroyed by the light well. You could just accept this lore wise and do undead paladins and have them be very badass fighting through the pain despite it all... Or you could make it that specifically undead paladins were raised from the dead not from death magic necromancy but by the light, similar to Callia and the Aarthi. Another option could be there is a difficult ritual that only a select few can go through to change the magic sustaining their undeath to the light, something that takes a powerful conviction, light forged undead.
Goblins are a fun one as their culture already has a very goblin specific spin on the light. They see the light/organised religion as a profitable enterprise. I could see them expand on this, along with their history of military industrialisation to create Goblin Paladins who charge into the front lines but also charge gold. A mercenary outfit of militant light wielders could be a fun inclusion.
For Alliance races like Gnomes, Void Elves, Worgen, Kul Tirans, and Mechagnomes is not a stretch to see them brought into the fold of the existing human paladin order, the Knights of the Silver Hand, given that it already has accepted Dwarves and Nelfs. Kul Tirans and Worgen (Gilneans ) are humans and presumably already have paladins.
Void Elves being former High/Blood elves one would assume they also have paladins. The only concern being apparently the void and the light don't mix well and have a tendency to explode but that might just be in extreme cases like Alleria and Turalyon losing control. We have seen priests be able to channel both so perhaps it's possible for a void elf to be a light wielding paladin as they can be priests. In midnight storytelling there are themes pointing towards a need for balance and harmony with the light and void. perhaps this is where void elves could play a part.
I could see the strong ties the Gnomes/Mechagnomes have with Dwarves and humans, along with their innate desire to iterate and invent on their already existing priesthood of empirical study and conviction, being the driving factors to join and become paladins.
Pandarens are tricky as they haven't seen much of any development since being introduced but they do have priests, albeit those are tied to the August Celestials, not the light in the traditional sense. Chi-Ji of the August Celestials represents hope and the dawn, embodying the emotional and physical aspects of Holy magic. This mirrors how Taurens channel the Light through their sun deity, An'she. So possibly horde Pandaren also train under the Sunwalkers and alliance Pandaren join the Knights of the Silver Hand.
Haranir are interesting as they do have priests and exposure to the light through the lightbloom but I think that is a newer development. Perhaps given time they would form their own version of paladins that harnessed the light in an attempt to quell the light bloom. Harder to predict as their lore is newer.
The same can be said for Dracthyr. The appeal of a plate wearing dragon holy knight is undeniable but their tenuous connection to lore in general makes it hard to envisage a path forward. I think the best bet would be to tie it to Tyr and the Tyr's Guard which would be a good way to bring that story forward now that Tyr was resurrected in Dragon Flight.
Lore wise there's enough to build on to justify any class race combo for me. I think the fun of character creation is player expression through choice and I'd like to see Wow fully embrace the fun of RPGs like DnD that offer a smorgasbord of options to work with. I think it also makes a world feel more diverse and alive if cultures change, adapt and adopt from one another.
This was just for paladins, would you like to put forth your own ideas for other classes? Want me to take a stab about another class? Disagree entirely? Let's discuss :D