Live Mixer for Sunday Keys includes eight different mixer patches so it's designed for a small number of instruments that you could run into it and mix live. Check out the demos on the product page and you can view the installation steps for an inside look at how it's designed.
Sunday Mix for Logic Pro X is designed for recording or post-production, but you can route audio through Sunday Mix’s audio tracks in real time for a livestream as long as your computer is up to the task and you have the needed i/o to do so and adjust for latency on the output side. You'll need some sort of broadcast software to get the audio output from Logic synced up with your video. Some features included in Sunday Mix are only for post production - like the kick and snare replacement. Others like the Sub Kick will work in real time. Check out the demos on the product page and you can view the installation steps for an inside look at how it's designed.