Concerning the part of the question, "How is this sequence timed and driven? Is it just aerodynamic forces, are there any control strings or is it something different?" there are passive methods and active methods.
Passive methods include that mentioned by @BobJacobsen, line drag, which slows the inflation by the time it takes to pull a line through a conduit in the canopy. Another is an "energy modulator", such as a rip-stitch tape shown in this JPL presentation on parachute deployment for the LDSD (Low-Density Supersonic Decelerator) tests, and this COSPAR presentation (abstract only) about reducing opening shock for scientific balloon flight termination.
Active methods often involve pyrotechnic line cutters, activated by an electric signal sent by a timing circuit, g-switch, or accelerometer feeding a pyro control system. I worked on the parachute for NASA's Genesis mission that returned samples of the solar wind to Earth, and it used pyrotechnic line cutters. Actually, it was not strictly a parachute, it was a parafoil, which unlike a parachute generates aerodynamic lift, not just drag. Unfortunately, due to a g-switch installed upside-down on the re-entry vehicle, the parafoil never got the command to deploy. Had it deployed, it would have opened with "brake" lines (that pull down on the trailing edge of the parafoil) shortened to reduce the forward opening surge you get with a parafoil. After the initial opening transients had decayed pyro line cutters would have severed the short lines maintaining ~60-cm loops in the brake lines, letting those loops straighten and the brake lines go to full length, essentially "coming off the brakes". In helicopter drop tests of the parafoil that system worked very well. The folks at Pioneer Aerospace, who built the Genesis parafoil, spoke of using such pyro devices in a wide range of parafoil and parachute applications.