Hi!
Let's considere a LED. A single one, fed by an adjustable power supply 
(capable of of both constant current or constant voltage). LEDs are 
powered by a current (20 mA for the cheapest ones), that's how you can 
modify their brightness. But there is a minimum voltage so they can 
light up (Vf, forward voltage). Too low voltage, nothing will light up 
(NO conduction) and too high (leading to a too high current) will fry 
the LED.
So for a known and stable voltage, a series resistor is inserted:
R=(Vsupply-Vf)/I
If Vsupply is 12V on one end and 9V on the other end, you can either:
-change R in series with your LEDs along the entire path (requires 
-de-soldering of SMT components). High R close to the power supply, 
lower R away from it. Err... not easy.
-use larger wires (Rwire=rho*length/section). Errr... not easy.
-use "dual end feeding". Run a pair of electrician wires, as the ones 
used for a lamp(2.5 mm2?), from your power supply to the remote end. 
Near end and far end power cables will share load-> reduced current in 
each->lower loss->brighter remote LEDs. Easy and cheap !