How can these fluid dynamical smoke-ring phenomena be explained?

I currently don't have access to the paper, but this should meet your needs:

Vortex Ring Interactions (by Wakelin-Riley)
https://doi.org/10.1093/qjmam/49.2.287

Here is my interpretation: Consider two rings moving co-axially in the positive x-direction with their rings in the y-planes (ring A in front of ring B). The particles in each circle of the (fattened) ring circulate and force spreading of the air in front of it (see http://en.wikipedia.org/wiki/File:Vortex_ring.gif). Then

1. A's circulations will force B to spread and enlarge (visually look at the vectors in the vicinity of A). Then A moves through and this repeats, because B's circulation will now spread A.
2. Here ring A is moving in positive x-direction and B in negative x-direction. Their respective circulations will stretch each other and slow them down until they meet at rest and fade.

Here are more direct papers:
A Note on the Leapfroggeing Between Coaxial Vortex Rings at Low Reynolds Numbers (by Lim) https://doi.org/10.1063/1.869160
Interaction of Two Vortex Rings Moving along a Common Axis of Symmetry (by Oshima)
https://doi.org/10.1143/JPSJ.38.1159