If a device is connected to a Wi-Fi network via WPS, will it stay connected if the SSID of the network is changed at a later date?

They will need to be reconnected manually.

This isn't a theoretical question or answer, they will disconnect as soon as they miss enough beacon/management frames from the access point. This behavior will be the same across all 802.11 capable devices. It's not that the AP has disappeared when you change the SSID, but it's identifying information has changed from the client view. (BSSID + SSID combination) and their perceived signal strength to the access point will be zero, as there are no longer beacon/management frames being received from the client, this means no RSSI (i.e out of range).

IP Addresses or their DHCP leases have nothing to do with keeping a 802.11 connection alive. Wireless access points work at Layer-2 of the OSI model, or the "physical/link" layer-1 of the TCP model.

Most, (if not all?) Operating system wireless network profiles are based on the SSID. So the client devices will see the new SSID as a whole new network, and thus you will have to reconfigure the clients with the new network credentials.