I had same kind of problem with my '08 270. Riding downhill and getting top of some bigger steps caused the bike to run lean and even stalling, like the fuel was running out. First I checked the float height but is was ok. Then after being pretty p****d out with the whole machine I took the carb out (again) and took a proper look at the carb.
The Beta Keihin has only one air intake to the float chamber, as the exhaust pipe so close on the other side. The chamber air intake channel in the carb body is very near to the gas surface even when the bike near level, so when going down enough or after bigger steps when the front of the bike goes down again, the gas can begin to flow out through the chamber air intake channel and its hose, by the "siphon" effect (is that the correct word?). And having only one air channel, this causes vacuum to floatchamber and the bike will be running lean.
This was also clearly visible, after knowing what to watch, I went to test it and always when the bike began to run lean, there was some fuel flow in the chamber air intake pipe.
To solve the problem, I suppose opening the other air intake could help (as it seems that all other bikes with Keihin have both of them in use).
I simply fitted longer hose to the air intake and routed it to go loop high enough and down again where it originally goes. "High enough" was about the height of the gas tap.
It has been perfect since.