How are long runs of LED strip controlled from a single switch/tuner?
This is actually super-easy. You'll laugh.

See your drawing where the one controller feeds several strips and also the amplifier? The controller's output is a PWM signal.  You can split that to any number of amplifiers. 

So you have as many instances as you need of 

                              /-&gt; Strip
 Power Supply ---- Amplifier +--&gt; Strip 
                       |      \-&gt; Strip
                       |
                       |
                       |
                       |
   one controller -----+------- (another PS/amp/strip group)
                       |
                       |
                       |
                       |      /-&gt; Strip
 Power Supply ---- Amplifier +--&gt; Strip 
                       |      \-&gt; Strip
                       |
                       |
                       |      /-&gt; Strip
 Power Supply ---- Amplifier +--&gt; Strip 
                       |      \-&gt; Strip
                       |
                       |
                       |      /-&gt; Strip
 Power Supply ---- Amplifier +--&gt; Strip 
                       |      \-&gt; Strip
                       |
                 etc. etc. etc. 


And each amplifier gets (not shown) the 4 wires coming off the controller. 


Note that in this drawing I split off the strips, not daisy-chain the strips nose to tail.  Even with 24V strips, the voltage drop becomes noticeable, and worse, chaining too many LED strips can melt the thin little PCB traces.  They're not superconductors :)   24V at least can make it five metres without visible voltage drop.


That illustration isn't showing three 5-metre LED strips daisy chained. It's showing three 2-inch segments that have not been cut apart. 

Note that I have the LED controller itself feeding no LED strips directly. They're all through amplifiers.  I find that in practice, controllers have limited capacity and it's not worth it for the few strips they can drive.


I also want the freedom to put the controller anywhere, not near strips.  
I also want the cleanest possible PWM signal from the controller since it gets distributed all over the place to the various widely dispersed amplifiers. 
I am concerned that "lag" in the amplifiers will cause amplifier-driven LED strips to look different than controller-driven ones.  So I want them all driven by amplifiers. 

Also note I proscribe multiple power supplies.  That's because voltage drop is a real bugaboo for low-voltage + high current stuff, and so you don't want to carry it long distances if you can avoid it.  Better to tap locally available AC mains with additional power supplies.  


So what's the deal with the signal coming out of the controller?  It is both LED power and also signal.  The controller's job is to pulse its red/green/blue outputs, hundreds of times a second; the duty cycle decides the red/green/blue intensity. So it is a signal, which the amplifiers input, and they pulse their outputs to match.  But it's also power: the output transistors are big enough to power a couple of LED strips, which allows small projects to be built without the need for amplifiers.