Linking genetic sex to a set of climate adaptations

YES, BUT...

You could make this be the way you want, if that is what you want. I'd disagree that everything CAN'T be a generalist in a world where nothing is stable, but it's your world. If the genes for surviving in each biome exist in all individuals, then you can trigger them whenever you want. A guy wandering into the mountains can have his lungs change to handle the lower oxygen, then change back when he leaves. Otherwise, the children become specialists and die come the next transformation. With this level of change, you would lose a lot of adults with each cycle, and offspring would need to be able to fend for themselves. It works for simple organisms with fast reproductive cycles, but not for more complex ones.

Simple organisms have lots of choices to deal with your environment. A lot of your species may create some sort of cyst or spore that is highly resistant to damage, and may survive until conditions change. There are some fish that survive lakes drying up and hatch when water returns. They could encyst, lay resistant eggs. or have eggs carried on the feet of migratory birds. They certainly could migrate a lot, and if you want specialists, they would need to in order to find compatible ecosystems. For plants, the seeds of every ecosystem would lie buried until the appropriate environmental conditions occurred. Big trees are unlikely, but their reproduction might DEPEND on environmental transformation in the same way some pines need fire to release seeds.

My suggestion for what you want would be adult transformation. If the SAME 2/3rds of your world were constantly changing, then you could skip reproduction as the change phase and have animals transform in a chrysalis when the environment changed, so the carni-bear transforms into the herbi-boar as food sources change. This way, the same individuals can survive multiple environments and still be viable. In this scenario, your organism reproduces in the middle of the stable years when food supplies are stable. They also build up fat reserves in the good years to fuel transformation and survive the rough years.

Your stable 1/3rd of the world could have specialists, but they wouldn't have a strong drive to colonize the unstable regions (except transiently). Herds could spread into newly compatible areas, but they would need to migrate back and cause overpopulation every time the environment changed. between each land change, the regions would be devastated and take time to settle back down. I'm guessing your on-site transformative specialists would outcompete migratory species. I think this is your desired outcome.