I would imagine that the 2000 to 3000 years ago (I wonder if they meant 3000 years BC, since the cores go back 6000 years), the rainforests were just in a different place. Say SW North America which is now desert. As the climate changes in various places, the species (plants, animals, birds, etc.) just follow.
I recall reading some National Geographic article years ago. One article was about some animal that is being more and more rare in the U.S. due to Global Warming, and something needs to be done, etc. (The article did bury the fact that they are just heading north into Canada).
In the same mag, another article was shouting with joy the sighting of some rare bird, the first time it has been seen in the states for 200 years, and people from all over were coming to see it. (It’s usual range was a few hundred miles south of the border in Mexico. No mention of global warming).
I was watching on PBS within the last month, the reason the rainforests in the Amazon exist is due to the dust kicked up in the Sahara desert in Africa that travels around the world, creates rainclouds and drops.
We know that several thousand years ago the Sahara was no desert. And now we see that the Amazon was no rainforest around the same time.
Climate changes all on it’s own. Apparently alot.
You answered my question; 2,000-3,000 years is not enough time not even probably 100,000 years. So there must have been continuous rainforest environments around for a long time. The forest area can grow and shrink.