The best way to troubleshooting the issue is vpxa and hostd logs on problem ESXi host and Web Client logs.
Please also check out Windows event logs if Web Client is installed on a Windows server.
BTW, you say Web Client has problem but vSphere Client is fine, I would say you may not able to notice the disconnect/reconnect on vSphere Client if it's flapping very fast. ARP conflic or IP conflict also can cause similar problem.