Hello @tmorris1 
Wi-Fi 6 is backward compatible with previous 802.11a/b/g/n/ac Wi-Fi standards, and kasa device supports 802.11b/g/n protocol (some supports 802.11ac), thus smart devices should work with a Wi-Fi 6 AP/Router.  One thing to note that not all the kasa devices supports WPA3, if router uses WPA3,  Kasa might have some trouble connecting to WiFi.

As for the instability problem, 1>  it is suggested to ensure kasa devices get strong signal: 

If router and AP/ extender use the same SSID,  device may roam to a different AP when signal gets weak or unstable: 

( -40~-50dBm very good signal; -50 ~-70dBm good to average;  less than -70dBm: weak signal,  connection may drop and need to locate kasa devices closer to router to ensure performance) 

2> you could use a different SSID and a different 2.4Ghz channel for Wifi6 AP and see if that help with stability. 

If the concern persists,  May I know the model of router, and the Wi-Fi 6 AP, are they a mesh system?

With WiFi6 disabled, does smart device work stable as before? How about the WiFi led status on smart device when issue happens, does it go red or stay green?