We've had flies, wasps, bees, crane flies, damsel flies, frogs and, obviously, butterflies and moths.
Most of these can be easily sorted - flies get swatted, bees are easily steered out an open window, and even the frog was reasonably easy to catch and release - but butterflies refuse to cooperate, fluttering around the glass ceiling or getting into the corners of the roof where we cannot reach them, and then dying from the heat.
Once they're dead, we can retrieve them with the tube on our Dyson, and that inspired a simple solution.
The tube of our Dyson is also the handle, with a plastic grid over the tube at the top*. You're supposed to take the handle off and reverse it for normal "tube mode", but it also works the wrong way round, and that inspired the solution.
I drape a piece of fabric over the grid to spread the air flow (usually a dish towel, but I've also used a sock), switch on, and then prod at the butterfly. The air-flow quickly and gently traps the butterfly's wings against the cloth. Hold the end of the tube out the door or window, cut the power and the butterfly flies safely away.
This method will work well for butterflies and moths, but not for strong fliers like bees and dragonflies - they get caught by the air-flow, but are strong enough to walk across the cloth until they are out of the air-flow and then fly free. I haven't tried on pests. For those, I leave off the cloth and open the grid...
*According to the salesman, the grid is there to stop the Dyson sucking out the eyeballs of small children. How could we not buy it after that??
(Thanks to Conker-X for taking the photos)