I have heard stories about audio lag giving problems with lip sync.
That seems plausible, but not definite.
I don't know what the current situation is, but there were fears that the 'unregulated' radio spectrum used for analogue radio microphones in theatres and such was going to be turned over to some other use, so live stage performances would be forced to use digital wireless transmission - with the associated time synchronisation problems*.
* Only encoded data streams are licensed in the relevant frequency bands, which require frame-based encoding - so a frames-worth of analogue samples have to be accumulated before they can be processed, and however fast the processing and transmission is there cannot be less than one frame time delay between the input waveform and the output waveform, typically 64ms. This is irrelevant for one-way distribution, but in live performance a singer will be receiving back a delayed version of their own voice and it won't be in sync with the backing.
Unless the Bluetooth transmits the raw digital audio data stream, there will be an encoding of the analogue audio (or its decoded digital equivalent), transmission via Bluetooth, and then conversion back to analogue audio for the actual headphone. Even if only simple PCM is used, there will be a delay... but the delay could be compatible with the flight time of sound from source to ear.