[nrjames]

There are many different ways to move objects through a flow field. Some people use the vectors to accelerate particles, others as instructions for instantaneous rotation, etc. The crossing lines aren't necessarily following the flow field in those images, but likely are just using vectors as an input to another set of instructions.

[mkl]

Yes, okay, but the algorithm described just before that treats the vectors as tangents to the path. The pseudocode does a complicated-seeming grid lookup instead of just evaluating the function, but apart from that it's just following the vectors as tangents.