Now when you say "dumb switches", are we talking home-quality equipment that you can't manage at all? If they have any remote management at all they will usually tell you what port they see a given MAC address on.
Failing in that, go and buy an old Cisco 2950/3550 switch for less than $100 and run all of your dumb switches off of that which will allow you to find the misbehaving device pretty quickly.
Somewhat dangerous but workable: download a packet generator from https://wiki.wireshark.org/Tools#Traffic_generators and hammer the MAC address. The switch handling the MAC should light up its port like a spot light. All of the switch interconnect ports will light up as well so identify those before you decide where the device is plugged in.
Last ditch method - plug your WireShark into one of the switches. Turn other switches off until you no longer see the MAC. When you think you know what switch the device is in plug your WireShark into that switch and start disconnecting patch cables until you've eliminated the other MACs.