"The branches of the world ride their own streams" comes from The Story of Warning. The original text of this sentence is "each branch in the world rides its own stream, each has its own ferry, each has its own boat, everything is life, and it is not up to people at all." Each ferry goes out and each boat comes back, which means that each has its own destination.
This sentence means that everyone has his own wheel of destiny. At different intersections, there will be different encounters, and everyone has their own karma and opportunities.