We started with FaceBook Live via an iPad. That was less than ideal, but it works - audio was the biggest issue; people near the iPad were often more present than content from the service.
We now use a camcorder (with decent optical zoom a member donated) with a black magic HDMI to thunderbolt converter to a Mac Mini running OBS studio. OBS studio handles all the transcoding to send to Facebook - it works a treat (other than having to deal with Facebook - not a fan but it works and is free). We also have a send from our soundboard to the Mini so the audio is super clean and free of audience noise (or us talking in the booth as sometimes happened with the iPad was in the booth with us). OBS Studio handles all that. We tried a few different Windows PCs and kept having random issues, especially after Windows updates - since moving it to the Mac it’s been flawless.
We also use the awkwardly named Newtek NDI Scan Converter, part of the free NDI tools, to send a stream of what Pro Presenter displays at the front of our auditorium to OBS studio and in OBS Studio we have it as a picture in picture in the corner so there’s a clear version of the slides along with a view of the stage. We also save a local recording if we want to have something better than what Facebook saves automatically. I just wish I could stream to Facebook and also stream RTSP from OBS studio to also have that view in other parts of the building - for the nursery workers, people in the kitchen working on prep, etc. There may be a way to do it, I just haven’t had time to dig and see.
BTW - NDI is awesome. I’m never buying anything but cameras that can speak NDI natively. Analog/HDMI is dead to me (hooray!) Thankfully NDI seems to be taking off, there are quite a few - especially in the PTZ (point/zoom) side of things now and the premiums they used to command vs other cameras are slowly disappearing. If they just had an inexpensive device that could display an NDIS stream - like a chrome or fire stick but for NDI - then I could ditch RTSP for internal use too.