What do you mean by outside network? It is outside your firewall connected directly to the Internet? If so, Why? There's really no reason for that, as it would be able to download files as quickly from behind your firewall. And you can set up your router to port forward and use MediaBrowser from outside your home network.
By outside network I mean it is on a network away from my. Serviio is on a server that has 1 gigabit connection to internet, and my home network has only 10 Mbps. It is quite a pain to download a hi-def movie at my home, so I'm downloading it to serviio and then watching it via mediabrowser with my PC. Now I want to make my Raspberry PI to play those files, so I could connect it to my TV.
If you mean it's on a network away from yours, i.e. in some other location, then the only option is likely to set up a VPN (Virtual Private Network) between the two networks. However, even with a VPN there are routing issues that could cause difficulty.
VPN might be an option actually! But that is quite difficult approach i don't really like. Since DLNA uses multicast for server announcements (not 100% sure, correct me if I'm wrong), and native VPN does not support multicast, i have to go with GRE over IPsec, so I have to put two routers on both sides, which would support that kind of VPN and then route that traffic through. I guess I will have to go that way if there will be no other solution.
DLNA will not transport on the Internet. The devices must be on the same network, or a network with routing set up between them with the proper configuration. I have no idea of Kodi would support any other type of connection. If it has a web browser built into it, you might be able to use MediaBrowser inside Kodi.
I'm not sure about web browser in Kodi either, haven't looked yet, although, i guess VPN is better idea
Raspberry doesn't have powerfull CPU, even Kodi interface gives it hard (up to 100% utilization), so I guess it is gonna be very slow. So I would like to make it as simple as possible. Write some plugin, which would act like backend for serviio to pull video files (instead of mediabrowser).
Raspberry Pi is a very tiny PC, you can put it in pocket, so I want to make it work everywhere I go. Just plug it in, and it will play.