RGB - Don't think it had any.
Yellow - The cruelty of humankind represented and argued for by Lance, and the good of humankind represented and argued for by Yellow.
GSC - Personal identity, most clearly in the "what are their special skills" scene, and preceded by Gold identifying his unique perspective of what Pokemon to make him one of the then-five Pokedex Owners.
RS - 1, love. Both Norman's fatherly love for Ruby leading him to take the fall before the Association, the protags' obvious mention, and Courtney spelling it out to him. 2, harmony between opposing elements, and the need for an outside mediator/balancer. Both the human and Pokemon red-blue pairs, really, with a lot of people playing the Rayquaza role for Ruby & Sapphire.
FRLG - Family. Both Green, Blue, and Giovanni are driven by theirs.
Emerald - 1, Love, as a continuation of RS's theme. Emerald's proclamation of the kind of people he cares about is the climax of his character growth during it. 2, family like FRLG. The arc ends on Emerald having a found family, many new friends and non-blood siblings, spelled out by the Latis.
DPP - Again a clash between the good and bad of humanity, Diamond and Pearl and Platinum explicitly representing secular virtues.