Best answer by Airgetlam
In my experience, an input lag generally indicates that there's a problem with the wifi signal getting through. I'd definitely take a look at the wifi interference FAQ, but I'm actually wondering if there's a chance that your phone and the new CONNECT:AMP are sharing the same IP address, and it's the bouncing between the two that's causing a slight delay that you're experiencing.
I'd recommend that you submit a system diagnostic within 10 minutes of replicating this issue, and contact Sonos to discuss it. There may be information included in the diagnostic that will help Sonos pinpoint the issue and help you find a solution.
If you're interested in doing some testing on your own first, I'd recommend that you unplug from power all your Sonos devices. Then reboot your router. Finally, starting with the Sonos device that's plugged in to your router, power it back up. Wait until that one is powered up before moving on to the next Sonos device, and plug it in. Repeat for as many Sonos devices that you have.
That would be the quick and dirty way of resetting the DHCP reservations in your router. Not 100%, but pretty much all your Sonos devices should get new IP addresses that way. If you wanted to do a similar thing in a more permanent manner, you could set up reserved IP addresses in your router's reservation table, but that tends to require figuring out how to do that with whichever router you have, sometimes an extra step.