I'm trying a "Swift swim" gameplay in radical red, but I wondered if is it ok to use kyogre's primordial sea in replace of drizzle. And of course I made every battle into a double battle type.
According to these sources yes, they do activate
Desolate Land activates as a Groudon equipped with the Red Orb is sent to the battlefield and triggers its Primal Reversion. Much like regular Drought, the harsh sunlight will increase the power of Fire-type moves by 1.5x while triggering the usual effects on moves and abilities like Solar Beam or Chlorophyll.
Opposing to Desolate Land is Kyogre’s Primordial Sea, which works exactly like Groudon’s ability. The heavy rain will activate whenever a Kyogre holding the Blue Orb item switches in, and will boost all Water-type moves by 1.5x while triggering the usual effects on moves and abilities like Swift Swim and Dry Skin.
Yea abilities like solar power and chlorophyll works in Desolate Land while Swift Swim and Rain dish work under Primordial Sea.