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The green network with the NASLite is 192.168.1.x and the blue, which I'm trying to come through is 192.168.2.x
Right, so you have 2 networks.
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the router should forward to the new network
If you're going thru a router you definately have 2 networks.
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I can also set the subnet mask of the NASLite to 255.255.0.0 which should read both networks.
Except NASLite knows you're trying to cheat.
When it gets a packet from your blue network it sees a network address of 192.168.2.x with a netmask of 255.255.255.0. When it tries to reply to that address the netmask tells the NASLite box the destination PC is on a different network and NASLite doesn't know how to get there.
What you need is a default gateway setting on the NASLite box so it knows to send all packets destined for other networks to the router. Unfortunately this is not provided in version 1.x for security reasons. It is promised in version 2.
Other network appliances will likely have a way to provide a default gateway address (your router), so they work accross networks.