GENERAL
I know parts of these have been how it works in the past, but I wanted to include it to expand across the full idea.
The speed stat would be used to determine how fast you move. Higher the stat, faster the movement. The scaling would be based on gameplay testing.
The attack or special attack stat, depending on the mount type, would determine jump height. If the mount is a physical mover (for example, Rapidash, Arcanine, or Tauros running), the jump would be based on attack/"muscles", while if it is a special mover (say, a flying mount propelling up with wind currents), it would be based off the special attack.
An extra element would be the ability for some mounts to have some secondary function outside of being able to be mounts, for example having Ursaluna and Stoutland (as well as others, perhaps Boltund) being able to function as an item finder to an extent.
LAND
Honestly, land travel is pretty good as it stands. Being essentially horse movement fits perfectly to riding Pokémon like Rapidash and Arcanine. What they are currently missing is the jump height of horses.
On an alternate concept, perhaps a mechanic akin to horse breeding in vanilla could become a thing, to allow for horse racing to be even more interesting with Pixelmon? Having speed, health, and jump be something that can be bred down (although writing that out, that just sounds like IVs doesn't it)
WATER
Water traversal is broken in 9.0.1, but ignoring that issue, I think there is two ways to break up water mounts, to allow for some interesting changes.
SURF
Surf would be riding across the surface of the water, akin to riding in a boat, but faster (depending on speed stat). Some Pokémon, such as Lapras, would only get this traversal method. Some Pokémon, such as Basculegion and Sharpedo, would also be able to jump out from the water, Basculegion having the double jump functionality present in Pokémon Legends Arceus.
DIVE
While surfing, pressing the Descend keybind would allow you to dive (if RequireHMs was enabled, you would need to know Dive). In terms of movement, I don't really have any ideas, but player swimming could be a decent jumping off concept? Looking up and down to move up and down, basically always moving towards the cursor.
LAVA
Based upon both the 1.16 mob the Strider, as well as 9Tales's great suggestion on this same concept, Lava mounts are something that definitely should exist within Pixelmon. As said by 9Tales, Nether traversal is currently not very easy, with flying not being super consistent or accurate, and land travel being difficult at times.
The ability to walk or run across the lava via a Hisuian Arcanine would be extremely fun, and expand the gameplay within the nether immensely.
FLIGHT
When thinking about flying mounts, a few ideas come to mind. Each have benefits and downsides, but I think a combination of the concepts across different Pokémon could work out.
GLIDER
Essentially Elytra flight without the fireworks, these Pokémon would be able to descend from places gracefully, and be able to maintain some air speed and distance from accurate movements. I would picture Pokémon such as Garchomp in this category.
SKY-SWIM
As the category name suggests, these mounts would somewhat function like swimming (or the Dive mount ideas from earlier), but in the air. Fast and simple maneuverability, but maybe not as fast movement as other categories overall.
FLAPPING
Pokémon who flap their wings would be this type, allowing for hovering, and bird-like traversal
SOAR
Taken from OmegaRuby/AlphaSapphire, and arguably Let's Go Pikavee, just being able to sit in the air, ascend when holding Jump, descend when holding Descend.
CLIMBING
Being able to scale vertical surfaces is honestly a feature Minecraft itself lacks, having the Spider be essentially the only mob set with that ability. However, with Pokémon, we know of atleast one species which should be able to do it, if not more via the Rock Climb HM from Generation 4. Sneasler and the HMers should be able to latch onto a wall, and climb it as if it's a ladder/vine.
That's roughly everything I can think of at this time, I hope that as 1.16.5 Pixelmon is developed, mounts will get better, because as of 9.0.1, they can be a bit finnicky