Travel from Switzerland to Milan, Italy

With a Swiss Travel Pass, you are only covered within Switzerland, with some exceptions. Going from Leuk to Milan, the STP will cover you as far as Domodossola, Italy. From Zurich to Milan, the STP covers you as far as Chiasso, Switzerland (just before Como, Italy).

So you just show your pass at the ticket desk, and they will charge you for the Italian portion of the journey.

Eurocity is a type of train. It is not the train operator. More than likely it's either Swiss Rail or Trenitalia and you can buy your ticket from either.

However, if you are only buying a ticket for the Italy portion, I'd suggest buying the ticket from Trenitalia who operates the train. But double check that they are operating the train.

You buy the ticket anytime prior to travel. It doesn't have to be from the last train station coverred on your rail pass. You could even buy it online.

What you can do is this:

Select the train on www.sbb.ch, and start the purchase process. Enter that you have a "GA" as a reduction. GA stands for "Generalabonnement", and it is a pass that offers pretty much the same free travel as the STP. This is what Swiss commuters get.
(GA is for locals, STP for tourists, get it?)
The system will then issue a ticket only for the part that is not covered by the pass, and will give you the needed reservations for the EC trains. Doing like this ensures you have a reservation starting in Brig.

The EC trains on that route are operated by SBB till Domodossola, and by Trenitalia beyond that. You will notice that the crews change there. The actual train used can be either an SBB owned set, or a Trenitalia set. The only difference between them is the colour...

Thank you so much everyone!

The regional rail service for the Milan area is Trenord. Their website is: https://www.trenord.it/en. It appears the train rail line from Brig to Milan runs through Stresa. Back in September, I took Trenord from Milan to Stresa but had to transfer to a bus at Arona (no big deal) to get the rest of the way because they were doing track maintenance. Hopefully it's completed so you can ride the train the entire trip.

Has anyone used the Trainline app to purchase tickets going on this route?

I'm doing the a similar trip (Lauterbrunnen to Milan) and am using the Berner Oberland Pass + Swiss Half Fare Card while traveling around Interlaken. I was going to use the Trainline app to get our tickets from Lauterbrunnen to Milan but figured I'd hold off on buying any tickets until a day or two before we leave for Milan. We have a loose itinerary and I'm not sure when we'd want to leave Lauterbrunnen just yet.

If you have a BORP and Half Fare Card, use www.sbb.ch. Much easier than Trainline. Buy at a Swiss train station and they will take care of you.

the BORP will get you to Domodossola, but only via the mountain route, not the base tunnel. So change to the RE in Spiez. (Enter via Kandersteg in www.sbb.ch to get this route)

Then just buy a ticket Domodossola - Milano on Trenitalia. Easy.