It's not so kind to leave them sitting and not eating at this time of year either - and yes, I've got one or two silly things trying to be broody too
Mine will sit and hatch wood shavings
when I've removed the eggs.
You need to 'cool them down', literally - which is how cold water works, but may not at this time of year unless you can keep them somewhere to dry off. It's the brood patch, on the chest and tummy that needs to cool. Wire cages are used because they allow air to circulate and so cool off - they work more slowly than the cold water method, so i don't use them in preference in the summer, but might be better at this time of year - or maybe just sitting them for a minute in the snow!
Actually, I think snow is what i shall do as it won't leave them too wet but will work quickly and so allow them back with the rest of the flock quickly.