laying is interesting, I don't think layers can encourage/induce laying-only support it. I think giving them extra protein through the moult is a god idea but most growers is not much higher in protein (if at all) than layers if you look at the bags. I don't run mine under lights either in winter as don't like the idea of 'forcing' them to lay either. However, thinking about it-birds on the equator would get 12 hour days and therefore presumably lay more consistently throughout the year-unlike ours in the Northern hemisphere.
It would be interesting ( cos I'm a nerd) to find out what fertility is like in an environment that doesn't have much in the way of seasons. Chickens after all are jungle fowl and from a more southern latitude than this. No matter what we breed them to look like, genetically they won't be far from jungle fowl. afaik brown eggs lose pigment over the laying season (so what happens to brown egg layers on the equator-do they never lay as brown an egg than those up here?), the shortening of days has a profound effect on animals so I'm not surprised that fertility is down this time of year. so what I'm saying is that the fertility and depth of colour in brown eggs is coincidental rather than causal.
from what I've seen on muscovy pages is that further south, they hatch and breed year round.