I did it like this:
Start by making four unique FOFs, like you already have.
I chose to make then 64 FRACUNITs tall as to fit the Star Light platform texture lol
Give their control sectors unique tags as well - For example, I chose 1 to 4 for the sectors that host the FOFs, and 5 to 8 for each FOF control sector.
Then, to all control sectors, give it an unique trigger tag. I chose 9:
Once that's all set and done, it's time to make the FOFs move.
Draw a segmented "control" sector right below each control sector. You don't have to do this, but I did it because it's organized.
Set one of the linedefs to use action 300, set its tag to the trigger tag you set earlier (9 in my case) and set its trigger type to Once (so it doesn't run again afterwards).
For each linedef clockwise to the previous one, set them to action 405 (Move Planes by Set Distance) so they can actually move.
Set their target sector tag to the corresponding tag used by a control sector (5 to 8), both planes, a negative distance gradually smaller (it looks better if you use the height of the FOF for this. For example, my FOF is 64 FRACUNITs tall, so -64, -128, -192, -256) and a speed gradually larger (I just picked a random number, 16, then went with it: 16, 32, 48, 64).
Save, and the result should be the same as in the video.
In Sonic 1 they have a slight delay of like 1 second before they actually start moving down, so maybe you'd like to change the executor delay in the trigger linedef.
Attached is the wad I made so you can give it a peek.