There is no "VLAN process" that separates things or packets. There is no magic, you design and implement the network; define VLANs and always associate one subnet to that VLAN.
I have never used multiple subnets and just know if they're not the same , devices don't communicate.
YES, different subnets do not communicate with one another... however, that is what you are attempting to do, isolate devices from one another. Your network has been a simple/flat network on one subnet... you are changing to a multiple subnet environment.
If you were to maintain three unique and three dedicated physical (flat) networks and have them each using a different subnet you would have to buy a lot more networking switches and run additional cabling. Instead, using an Layer 3 (L3) switch you have one physical network that you can logically separate into three or more multiple networks.
Routers have the ability to forward (link) traffic from one subnet to another when allowed/defined to do so. Your ASUS router connects the WAN (internet public IP address) to you LAN (private IP address). These two IPs are not on the same subnet but do communicate via the router. A router is a device that connects two or more packet-switched networks or subnetworks.
A L3 switch is a special network device that has the functionality of a router (L3) and a switch (Layer 2) combined into one chassis. Just like your router that connects WAN to LAN, a L3 switch can provide communications (connections) between different subnets; i.e. VLANs.
When you define a VLAN (aka subnet) on a L3 you are defining a logically new/separate/unique network path. As you define which ports are assigned to that VLAN you are creating a logical switch connection for the device you connect to it. When a device is requires to be a member of multiple VLANs (multiple subnets) each NIC must be assigned from the IP subnet range of which the VLAN has been defined.
These are all manual steps that must be physically managed. Simply creating VLANs on an L3 switch, assigning ports to VLANS, and installing multiple NICs (on the same subnet) does not auto-magically make a valid network. I can see how the L3 switches are going nuts trying to make sense of the current environment.
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You should expand your knowledge from simple (flat, single subnet networks) to more advanced multiple subnet networks. A brief description of the pieces (layers) of network communication is below. I recommend further research/reading to improve your understanding of each of these layers and how they all work together.
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The Open Systems Interconnection model is a standard reference model that describes how applications communicate over a network. The OSI model gives us a visual design of how each of the seven communication layers is built atop another. It starts with the physical cabling and continues all the way to the application that is trying to communicate with other devices on the network.
There are seven layers in the OSI model:
Layer 1 (Physical)
Layer 2 (Data-Link)
Layer 3 (Network)
Layer 4 (Transport)
Layer 5 (Session)
Layer 6 (Presentation)
Layer 7 (Application)
