How I would rule:
1. Pixie makes a perception check to find the invisible PC to cast fly on him.
2. If the wizard saw the now-invisible PC disappear, or if the PC made noise after turning invisible, the wizard could make an arcana check to determine that the invisible PC is present and invisible. Depending on the circumstances, the wizard's passive arcana might be enough to pass this check.
3. If the wizard passed the arcana check, he or she could make a perception check to target the invisible PC with dispel magic, with advantage because the wand is visible.
3a. If the wizard failed the arcana check, he or she could make another arcana check on seeing the wand fly by. The DC for this check would be higher.
4. If the wizard successfully targeted the invisible PC with dispel magic, he would turn visible and lose his flight speed at the top of the stairs, since this is as far as his dash would have gotten him in one "round." (We're not in combat, as far as I know, but I'm thinking in rounds for the sake of timing.)
4a. If the wizard failed in targeting the invisible PC with dispel magic, the spell slot is wasted, and the invisible PC gets away.
4b. If the wizard failed the first arcana check but succeeded on the second, he or she would recognize what had happened but be too late to get the spell off. (Since the character was already invisible, I am reading "out of the other players' sight" to mean that the PC passes behind the wall or otherwise enters an area where the direct line needed for casting a spell is interrupted.)
4c. If the wizard failed both arcana checks or only did one and failed it, he or she would likely have come to believe that the wand was flying of its own accord, or that the invisible PC was inhabiting the wand or something else fun and funky. If the wizard tried to dispel the wand, the spell would be wasted.