Yes, tea leaves are a bit heavier than the tea (I mean the densities of course). But first of all this difference is not large. As long as you stir, there is some stationary velocity distribution in the cup, and the leaves are more at the walls than in the center. When you stop stiring, the tea motion is slowing down due to friction. The friction happens to be the strongest at the side wall and at the bottom. Due to this friction eddies appear. This can be seen by the following argumentation. When you stir the height of the tea liquid depends on the distance from the central axis of the cup - there is some bending of the upper free surface. Pick out a small box of tea at some distance from the center axis and at some depth. This box is rotating along a circle, so there must be some horizontal radial force which causes this. The force appears from the radial dependence of the pressure, and this radial dependence comes from the different height of the liquid for different distances from the center axis. If you think a bit, you can deduce that the height of the tea liquid depends quadratically on the distance from the axis center.
Now when you stop stiring, the height distribution in the liquid is starting to change. This causes eddies. Simply try to imagine how to redistribute the liquid, and you will see, that the eddies have the following properties: they carry the liquid upwards when close to the center axis, and downwards when at the side walls of the cup. Consequently when at the free surface the liquid is carried AWAY from the center axis, and when at the bottom the liquid is carried TOWARDS the center axis. Agree?
Now what about rivers? Well, we have to assume first that there is some inherent instability of a river bed with respect to some bendings due to erosion. What is then happening? Once a river bends, the transversal shape of the river bed is becoming asymmetric and correlates with the bending direction. This can be understood using our tea experiment. Indeed, since the river water is moving on a part of a big circle, it is just part of a huge tea glass we are stiring! So we know the orientation of eddies. At the bottom the eddies transport the liquid towards the center of the bending circle, and at the surface away from the center. Add the erosion, and you see that the river side which is closer to the circle center is getting mud and the opposite side is loosing it. Consequently the river bank closer to the circle center is shallow and the opposite bank is steep.
Now the law of BAER states a similar effect of shallow and steep river banks due to the Coriolis force. Namely in the Northern Hemisphere the right hand river banks are predominantly steep. In the Southern Hemisphere it is the left one.