I've been noticing this for a couple of months, but it took me till now to be really bothered by it as I was unable to sync content for a long trip.
The video should contain all the info you need: https://www.youtube.com/watch?v=JKrLaBXI6wY
But if you prefer a text version:
On my iPad Air 1, if I start syncing something, the internet/wifi on it just freezes/hangs. No websites load, and navigating around the Plex app just doesn't work. If I quit the Plex app, or stop/delete the sync, everything is back to normal.
This does not happen on newer iPads, such as the iPad Air 2 and 10.5" iPad Pro, or my iPhone 7.
It is not an issue with my Plex Server or network or anything like that, as it works on a newer iPad in the same exact conditions, same wifi network and everything (no 2.4GHz vs 5GHz difference or anything like that. No 802.1aac.).
I also tried doing it on a iPad Mini 2, which is basically a mini iPad Air, and it had the same exact issue.
But yeah, just watch the video. It tells you everything.
Oh, and in the video I use the current App Store version. Same problem with the beta version (that's in my dock.)