If I summon a pixie with Conjure Woodland Beings, and it casts Fly on me, can it continue to concentrate on keeping me flying if my concentration on Conjure Woodland Beings ends and it disappears?
It's been etablished before that distance, line of sight and even being on another plane don't affect the capacity of concentration, unless a DM rules that some circumstances might call for a concentration check. A sage advice article confirms that.
However, let's look at a part of the description of the spell Conjure Woodlands Beings :
The summoned creatures are friendly to you and your companions. Roll initiative for the summoned creatures as a group, which have their own turns. They obey any verbal commands that you issue to them (no action required by you). If you don't issue any commands to them, they defend themselves from hostile creatures, but otherwise take no actions. (PHB, p. 226, emphasis mine)
On the account of that last sentence, it can be safely assumed that the summoned creature isn't subject to verbal commands once it has disappeared, and thus is no more compelled to concentrate on a spell it was ordered to cast.
So while it would be technically possible (on the account that concentrating is not an action) for the creature to hold on to its concentration once dismissed, it will be up to the DM to decide if it has the will to do so, according to its nature and state of mind at the moment.
As far as the rules are concerned, the pixie can continue to concentrate on a spell after it disappears. The only things that break concentration are (PHB 203):
- Casting another speIl that requires concentration. [...]
- Taking damage. [...]
- Being incapacitated or killed. [...]
While the word "disappears" in the Conjure Woodland Beings spell is not really defined, it obviously isn't casting a spell, and it's used fairly consistently throughout the PHB to refer to summoned creatures returning to their native planes, without any suggestion that it is harmful in any way.