It seems unlikely that quickly growing roots could circumnavigate the planet in 9 years. But I have a way you could do it. Hopefully I'm not too far off what you are trying to achieve.
Imagine a huge root system that goes around the planet in a belt, draws it's nourishment from whatever lakes are around. As night falls on part of it, it's basically autumn, the trees lose their leaves, the local roots draw in what they can and go dormant while the trees eventually die and fertilise the earth while the roots lie waiting below the frost for the next spring/day. Mushrooms spring up etc,.
With the areas that are just lighting up the roots feel the change and start putting out shoots and growing very fast for 9 years. So you have a moving rain forest in a belt around the entire planet where it's daytime and a bit of the night. Thickest and biggest at the trailing edge, and more sparse and younger trees at the leading edge.
The rain forest fauna would have to move with it, but I can imagine all sorts of specialists that might thrive in the leading and trailing edges and among the dying trees in the night.
They can do the things like throw seeds everywhere, have symbiotic relationships with animals etc, as well. It helps them spread and compete.
You could even have the rain forests only appear in suitable places, with pine type trees in suitable environments for them, all depends what roots are waiting for the sun to warm them.
This is probably the only way you could have specialist rain forest fauna. Otherwise they'd all die waiting 9 years in the dark for the seeds to sprout, if everything moves then they can evolve to move with it. The reason I use roots instead of seeds, is the water supply problem. Roots can store a lot, and with 9 years you can move water for a lot of km if you're imaginative about it.
If the different species had all evolved to share water for instance, you could basically have a suitable environment for all plants with water permeating through for hundreds of kilometres over time. So grasslands, wetlands, the whole deal.
Basically you're just transporting the water underground instead of in the clouds.