I was getting a similar issue on my iPhone 6 within a few days of upgrading to iOS 10. Sometimes I would pick up my phone with over half my battery capacity showing, start replying to a text message and within 30 seconds I would get the “battery below 20% — do you want to turn on Low Power Mode”. Within a few seconds of that (no matter whether I said Yes/No) the phone would turn off completely. If I tried to immediately turn the phone back on I got the low battery icon indicating the phone did not have enough battery to boot up. However, if I waited 20 minutes, I could turn the phone on successfully, and the battery would usually show around 15%. If I then plugged in, it would immediately show around 40% and charging. Erratic to say the least. My partner’s 6S also had the same issue you describe.
I planned to wipe and DFU reinstall, but never got round to it and the phone started behaving properly again. In fact, it hasn’t turned itself off in a few weeks now. I have installed iOS 10.1.1 since these issues began so maybe there was a fix for the iP6 in one of these releases. At least there is some hope for my partner’s 6S in the upcoming 10.2 release.
I recall the last few major releases of iOS have either caused battery issues, affecting at least one model of iPhone, or complaints of a slow before a fix is issued. With that track record, you would have thought Apple might have tested their iOS releases a bit more.