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A ram air parachute requires airspeed to remain inflated, and its BAD, BAD news should it deflate. Any problems turning up wind down wind? Absolutely none, and nothing can turn quicker than one of those babies. They are certainly affected by changes in airspeed due to gusts and care has to be taken in such conditions because they will collapse
There won't be problems provided there is some airspeed as you say, and there usually is from below due to descending. Try an instant 180 "level" turn to downwind at 90 degree bank with a parachute and see what happens.... same effect as the gusts you mention: negative airspeed.
Ok I should've said "theoretical" turn. I wondered what the parachute example was trying to illustrate, since you said there are no problems turning up- or downwind and then say they are certainly affected by changes in airspeed! Of course they are, but providing they keep descending then they have some "flying" speed (in
parachute terms), so horizontal wind direction mostly wouldn't matter.
The issue here is that if turning sharply from upwind to downwind puts you below the airspeed for required lift for your aircraft (performance/capability), you go down at a steeper angle than you would turning upwind whilst trying to regain that speed, since you are now chasing the air you require. Downwind turns require more altitude to recover should lift be lost during/exiting the turn than upwind.