The agility 50 has a feature on the carburetor called a auto bystarter, which enriches the air/fuel mixture under cold start conditions to aid starting and running for the first few minutes of operation. The fuel bowl has a seperate fuel well, or tiny bowl that the auto bystarter mixing tube draws fuel from. This bowl is fed by a tiny orifice in the bottom which allows the tiny secondary bowl to fill with fuel to the level that the fuel rises to in the main bowl, though this takes quite awhile as the orifce is tiny. Upon cold starting the bystarter mixing tube draws nearly all of the fuel from the tiny bowl as a prime to get the engine started, after that the small mount of fuel coming through the orifice is mixed with air in the mixing tube and is fed through the air passage which bypasses the throttle slide, until the bystarter closes off the mixing tube, and air passageway a few minutes after startup.
The carburetor on the agility does not set level when the scooter is level. It is tilted to the side slightly, and angle up towards the rear of the scooter. When the scooter is on the incline with the engine off the secondary bowl could fill to a slightly higher level, and the level of fuel in the carb may rise to a higher level than normal as well which would also contribute to a richer prime and overall rich starting condition. (possible flooding)
While going down the ramp with the engine running, it is likely still on auto bystarter enrichment, and the carb vent tube air flow is more critical during that time and may temporarily have the inner passage to the bowl covered with fuel while at an incline, and this would cause the engine to stumble. Also the idle jet being towards the front of the carb, and while backing the scooter down an incline the idle jet may temporarily be above the fuel level in the bowl and suck air only causing engine to stumble.
Experiencing cold start problems if the scooter is level and unmoved could also indicate the carburetor needs cleaned and/or the auto bystarter and accompanying orifices and air passages are not functioning correctly.
Here is a great article about auto bystarters. Different brand and model, but same concept.
http://hondaspree.net/wiki/index.php5?title=Bystarter:_What_It_Is,_What_It_Does There may be some other factor causing your problem as well, but these were the first things that come to mind.