Can a flyer do running shot, end their movement next to a grounded fig and then do a ranged attack?

They certainly can. It's even more fun with figures like Captain Atom who have Running Shot and Super Strength- Running Shot adjacent to a Stealthed figure sitting on an object, steal the object out from under them, and then blast them with a ranged attack. Next turn, push to bash them with the object

Note though that they can only make the ranged attack against the figure they're basing.

Thanks for the quick reply. And yes, I was going to use your tactic: flying brick with ranged attack to-

1) with a poisoner as a passenger, do running shot and pick up an object along the way
2) end running shot next to 1 grounded fig, drop poisoner next to fig and do ranged attack, end turn
3) on my next turn, grounded fig takes poison and then gets object planted on his dome

To add to this if the figure you are attacking and in stealth and you have no means to see the stealthed figure then you cannot make a running shot attack.

This is a bit misleading- if the figure you're attacking is in Stealth solely because they're on an object, and you can pick up the object using Super Strength during the Running Shot movement, then you can target them just fine. You don't have to be able to see the figure at the beginning of the Running Shot, because Running Shot works thusly:
1) Give the figure a power action.
2) Move the figure up to half of its speed value.
3) Make a ranged combat action to shoot somebody.

This means that you can pick up the object with Super Strength during step 2, and so by the time you get to step 3 the figure is visible and able to be targeted.