Based off of what we know from PLL if we were to get the next 3 they would be:

Diamond Shiva/Amethyst Ramuh/Sapphire(?) Leviathan

They would all use their signature attack on summon for a flat high potency damage attack.

Their continuation abilities could be:
Shiva-Glacier Bash: Forms ice to crush enemy (like Titan's Mountain Buster)

Ramuh: Chaotic Strike: 2 part attack: 1) Send a lightning bolt towards a target followed by 2) a strong thunderstrike at targets position (Like Ifrit's Crimson Cyclone but without the gap close, or could include in part 1)

Leviathan: Aqua Breath: Form a puddle of raging water below the target, anyone in it takes damage over time. (Like Garuda's Slipstream)

Gem Shine/Brilliance will act same as they do normally, aspecting to recently used pseudo-primal's element.