Snot isn't brown, even in very ill goats. So someone has diarrhea she has gotten into. Most goats don't clean themselves with their mouth when they have diarrhea, so it may not be hers. Moms rarely clean kids of diarrhea either.
Lots more info. What is her temp? Do you worm? Is she normal wieght? A spring kid (April) would be nearing 100 pounds. Do you treat for cocci? She is in with her mom, is baby diarrhea a problem? Has it been in the past?
With the cold weather, goats who do not have access to a rumen full of hay or roughage will come down with pnemonia.
All goats have pasteurella bacteria in their noses, goats with good immunity and a belly full of roughage to burn all night to keep warm, will fend off this change in weather, the cold, the humidity. Does who are chilled, especially from rain or blowing wind, does who go to bed with empty rumens will chill, it lowers their immunity, the normal levels of bacteria in their noses grow, first you get snots, then fever, then a rattly chest, if you have not caught this by the time you see fever the goat will die without vet IV's and antitoxin.
Clean the kid up, take it's temp. Get back with us...my phone number is with every post. And it is not on my posts once kidding season starts

Vicki