OK , this is not using an arduino, but a very much older, dirt cheap chip - the 555, ( who's inventor died this week - RIP, he drew the artwork for the die by hand evidently ! )
You could use the CMOS 7555 to save power if you have 5000 of them ! 60 microAmps x 5000 = 300 mA plus the 20mA for the LEDs ( you could have 3 LEDs in series in each luminaire ) means less than 500mA needed for one or up to 10 dots chasing eachother .
You just need to hook some 4 wire telephone type cable between each lamp, ( I had them in the trees for a different project. )
For the number you are suggesting, I would have a pcb made, with an IDC connector to run some 4 way ribbon cable through ( or 8 way to share the DC ) That way there is no soldering to connect them.
The drawback is that the chase can only go one way !
Heres a bit of lateral thinking
The controller generates a pulse once every 3 seconds ( or whatever ) to start the chase, and sends a negative trigger pulse to the first LED unit ( luminaire ) this time is adjustable and could be set to leave a delay between one chase and the next, or sped up to have an LED coming on every 100 feet or whatever . You could have 3 or 4 light dots running up the drive, or just one.
The chase speed varies the time each LED stays on, and it is the LED switching off time that is passed to the next luminaire trigger, so you can have the light dot moving up the drive fast or slow.
Just an old fashioned idea, you could have an Arduino for the controller, it even comes with the right software - Blink ! 