In my mind, this default gateway IP address shouldn't be in Local site, but in the WLC side,in flexconnect local switching mode, this default gateway IP address should be in local site ,is it correct ?
Yes, if WLAN is configured with FlexConnect Local Switching, then clients default gateway should be the SVI defined on local site switch.
From the WLC perspective, WLAN can be mapped to a dynamic interface (or dummy interface) , this is not important for FlexConnect local switching client traffic.
How does the connected mode AP identify what kind of traffic should be transported by CAPWAP tunnel due to there is no mechanism such as ARP.
Each SSID needs to configure for FlexConnect local switching or Central Switching. If it is Local Switching, AP knows that traffic to locally terminated (not to CAPWAP back to WLC as of central switching).
HTH
Rasika
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