I know that in accretion disks, differential rotation means that inner parts move faster than outer parts. However, when angular momentum transport occurs (through viscosity, the magnetorotational instability, etc) the "inner parcel" of fluid loses energy, slows down, and moves to a lower orbit. But isn't the lower orbit moving faster? How do I reconcile the fact that lower orbits in accretion disks move more quickly with the fact that particles that slow down move to lower orbits?
The same way that in orbital mechanics, slowing down causes you to speed up.
For a simple example, consider a Hohmann transfer: A satellite in a high circular orbit makes a short retrograde rocket burn, causing it to slow down and move to an elliptical orbit, the periapsis of the elliptical orbit is closer to the planet than the circular orbit, but the apoapsis is the same as the circular orbit. Its speed at apoapsis is slower than the speed in the circular orbit, but its speed at periapsis is much faster.
It then makes a second retrograde burn at the periapsis of the elliptical orbit, this moves it to a circular orbit. Its speed in this orbit is slower than the speed at periapse in the elliptical orbit, but faster than the speed of the original circular orbit. Thus by slowing down twice, the satellite has sped up!
The same effect can be achieved by long gentle retrograde burns. The energy released from gravitational potential energy actually causes the satellite to speed up as it makes a retrograde burn. And the same happens as a result of atmospheric friction. As the ISS moves through the top of the atmosphere it experiences drag, causing it to move to a lower orbit and speed up.
The dynamics in an accretion disk are the same. Drag (as a result of viscosity etc) causes a parcel of orbiting gas to convert potential energy to kinetic energy and heat, the conversion of potential to kinetic energy means the parcel speeds up.