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It depends.
If you use an IP address based URL to access the AJAX Remote, it's pretty safe to say that this URL will change when you log onto another network. Usually, IPs are dynamically assigned to clients by WiFi hotspots and you may even get a different IP when logging into the same network.
If you use a Bonjour based URL and there's no other machine on the same network with the same Bonjour name, this bookmark should continue to work even when switching networks.
Best regards,
Felix Schwarz