In these plots, each point represents a particular orbit. The arrow at each point is a vector whose components are (dr/dt, diota/dt). [(M/mu) (dr/dt, M diota/dt) in physical units.]
Note that these results are quite preliminary, and some details might be a bit spotty right now. The key thing to take away is the relative magnitude of the evolution rate across the parameter space (eg, the arrows are much larger for small r than for large r).
For all the plots, the orbits with iota = 0 are most bound (co-rotating); iota monotically increases as the orbit becomes less bound.
Results for small r:
In all cases, the radius changes quite a bit faster than the inclination angle; this is a generic feature that I find for all orbits. Other features worthy of notice are how much faster the orbits evolve at small radius, and how much more quickly they evolve for fixed radius as the orbit becomes less bound (ie, as iota increases). (Note that at these radii, there is some angle iotamax < 180 degrees beyond which no stable orbit exists.)
Results for larger r:
The same general features exist here as above. The extremely quick evolution evident at r/M = 7 is an artifact of the way that I sample the stable orbits: in that case, I happen to pick parameters such that I am extremely close to the least stable bound orbit. In the other cases, the parameters are chosen so that I'm not quite so close; hence, no other orbits in this figure evolve quite so violently.
And still larger r:
In this plot, the evolution is so slow that I have multiplied the length of the arrows by 20 in order to make them visible ont. This plot mostly shows the same general features as seen in the other plots.
Fintan Ryan's work shows that the rate at which the inclination angle changes, at lowest order, is proportional to sin(iota). Thus, at lowest order, the greatest change in iota should occur at iota = 90 degrees.
In the following plots, I show that this is not the case for a/M = 0.8: the greatest change rate occurs at iota near 110 - 120 degrees. My guess is that this is due to terms going as (a/M)2; to test this guess, I am generating similar data for a/M = 0.1.
r/M = 8:
r/M = 10:
r/M = 15:
r/M = 20:
Unfortunately, I don't yet have any data for very large r. I expect that the peak will move to 90 degrees for r much larger than shown here. (Notice there is the trend of the peak moving from about 120 degrees to 110 degrees as r/M moves from 8 to 20.)