I brewed an IPA yesterday. All went well and it is bubbling away today. However, my friend's brewing fridge/fermenting chamber just freed up and he suggested putting my brew in there. It would obviously be better to ferment it in the fridge with temp control etc. I'd have to drive it to his house (5 mins away). Is it too late to move it? Will that just oxidise the beer? Or is it early enough that I won't have to worry about that yet?
Go ahead and move it. You won't oxidize the beer - the headspace is already filled with co2, and the yeast will scavenge any oxygen that does make it into the beer.
24 hours in, I don't think you have much to worry about. As mdma suggested, you still have active yeast that would gladly clean out any oxygen that finds its way into the beer.
That said, I would either move the fermenter very soon or not at all. The vast majority of your fermentation is going to happen in the first 2-3 days. That period of most active fermentation is when your beer is most sensitive to temperature. Once that very active fermentation has passed, it won't do much good to move the beer to your friend's temperature controlled fermentation chamber. The tail end of your fermentation just isn't going to have a sufficient flavor impact to make the move worthwhile.