How can I extract clips from a streaming video feed in near real-time?
This is fairly straightforward if you save to a streaming format like Flash Video (FLV) or MPEG-Transport Stream (MPEG-TS).

Let's say this is your full feed capture command

ffmpeg -i source {-transcoding options} localfile.flv


You can then run

ffmpeg -ss 6:12 -i localfile.flv -t 1:05 -c copy extract.mp4


Ideally, the stream should be past 7:17 or close to it when you run the latter command. Note that since the 2nd command doesn't transcode, inpoint seek isn't precise. The extract will start from the nearest keyframe earlier to the specified inpoint.