There are a number of different types of aurora, and the causes are still a very active area of research. A number of models have been proposed, and a fair amount of data has been collected as well. One of the challenges for gaining understanding is that we are lucky when we have even one spacecraft in the right place at the right time to measure what is going on above interesting auroral features that we see from the ground.
Some of the more rapid, small-scale dynamics of discrete auroral forms are often attributed to Alfven wave structures in the upper ionosphere (above 1000 km, say). These waves can have electric field components parallel to the geomagnetic field so can accelerate the electrons towards the Earth. The dynamics of the waves (incident from above) as they reflect from the upper ionosphere and/or interfere with each other and/or breakup due to instabilities can be complex. In this model the rapid change in the auroral arcs are a consequence of the parallel electric fields associated with those wave structures, not rapid movement of geomagnetic field lines. There have been measurements, including quite recently, of these kinds of wave structures above the aurora. The temporal and spatial scales associated with the waves correspond with relevant scales of auroral forms, and numerical simulations of these kinds of wave structures are also consistent.
Other types of observed auroral structures, such as auroral spirals, may be due to instabilities associated with the thin current sheets of downward-moving electrons. Some of the instabilities could create vortex-street like structures ("spirals") that you also see in other types of fluid motion.
I'm sure there are a number of other mechanisms as well, but I am not an expert in this field so don't know what-all is known. If you do a search on google scholar you should find many publications on this topic.
Jason