How to automatically turn camera to follow speaker?
High-end sports productions have cameras that can pan, tilt, and zoom their way to tracking a golf ball when driven 300+ yards off a tee.  Such systems are well beyond the budgets of typical user groups.  A system called SoloShot costs less than $1000, and might have the spatial resolution needed for your application.

Recording in 4K gives you a 2:1 punch-in for FullHD and around 4:1 or 5:1 punch-in for SD.  The problem with a 2:1 punch-in is that it only cuts your peripatetic speaker's travel by 2x, which means if they are dashing 18' from one side of a 12' screen to another, their apparent motion will still be as if walking 9' across the view, which is still distracting.  If you are OK producing with SD-ish video, a 5:1 punch-in will keep their motion to reasonably within frame.  The open source software Blender has motion-tracking software that can effectively steady a camera around the subject of interest (your speaker).  But you should check that it has the video codec support you need.  Blackmagic's DaVinci Resolve also has a tracker that can do this, but the free download version works only on Mac and PC, not Linux.

Another problem with using digital shot cropping is that at 24fps or 25fps, you might see lots of motion artifacts that seem strange: the speaker stays in the middle of the frame, but they are constantly blurring left or right, whereas the background is whipping back and forth but it's perfectly sharp otherwise.  Your eyes will tell you "that's backwards!".