The GN used their Mallets where they would do the most good. For the most part that meant the Cascade and Rocky Mountains, the Missabi ore trains or steep branch lines like to Butte MT.
The excellent book "Steam Locomotives of the Great Northern Railway", which I think is still available from the GNRHS Company Store, has a cover painting of R-2 2043 headed for Interbay on the Seattle waterfront north of King Street Station. The painting is based on a Warren McGee photo. I take that as proof that the R class, which were simple articulated engines NOT mallets, worked to Portland, at least for a while, probably after the diesels displaced them from the Rocky Mountains, which was their intended and initial service.
The worst grades on the GN were over the Cascade Mountains between Skykomish on the west and Leavenworth on the east. The crossing involved about 20 miles of 2.2% ascending grade in both directions. GN was very early (1906) user of the 2-6-6-2 mallet, class L-1. They were slow, riding on 55" drivers, and were initially used as pushers on the mountain. A consolidation plus an L-1 could take 1300 tons up the grade compared with the previous standard of two consolidations and 1050 tons with no increase in fuel and water consumption. The GN soon bought more to run one as road engine and one as pusher with trains of 1600 tons.
In 1907 the GN introduced the lighter L-2 class for use as a road engine between Leavenworth and Spokane, a route with long 1% grades in both directions. A second order was deployed between Whitefish and Cut Bank, which included the crossing of the Rocky Mountains at 1.8% eastward and 1% westward. Pushers were used on the 1.8% from Essex to Summit .
The GN was a late adopter of the Mikado 2-8-2, buying the first in 1911 and continuing thru 1920. The GN liked the power, speed and simplicity of the 2-8-2 and converted all of the class L engines to class O-5 and O-6 Mikados between 1921 and 1926.
In 1910 the GN bought ten 2-6-8-0 mallets, class M-1, again for the Cascades and again on 55" drivers. They were simpled starting in 1925, to become class M-2. They ended up on the Missabi Division, on the Butte division working the steeply graded line to Butte, and on the Spokane Division on the heavy traffic, and nortoriously crooked, branch between Hillyard and Kettle Falls. Most of them lasted into the 1950's on the Iron Range.
Class N-1 mallets, on 63 inch drivers, were introduced in 1912. Initial assignments were to the mountains, but the 63" drivers indicate that the company wanted a bit more speed capablilty. Each of the 25 engines developed 93,000# of tractive effort. Starting in 1924 they were rebuilt to simple engines, class N-2, with TE of 100,000#. In 1936 they were mostly on the Iron Range, with 6 on the Butte Division and two on the Kalispel. They handled 175 car ore trains of 15,750 tons, an indication of their power and the high quality engineering of the line from Kelly Lake to the ore docks.
The GN eliminated steam power over the Cascade Mountains in the early 1920's by electrifying first the 20 or so miles from Skykomish to Cascade Tunnel, followed by the 7.79 mile long new Cascade Tunnel with full scale electrification of the roughly 75 mile long line between Skypomish and Appleyard, about two miles east of Wenatchee.
The biggest and baddest GN steam engines were not Mallets, but simple articulated class R 2-8-8-2. The first four R-1 engines came from Baldwin in 1924. They were intended for the Rocky Mountain crossing. The R-1s made such an impression that the N-1s were simplified before any more R-1s were acquired. A second batch of ten R-1 engines, built to somewhat different specs than the BLW engines were built in the company shops in 1928. Sixteen Class R-2 engines, with larger fireboxes were built in the company shops 1929-1930. These engines produced more tractive effort than any other 2-8-8-2, 2-8-8-4 or 4-8-8-4.
Finally, in the 1930's the N-2 engines were all rebuilt to new class N-3 with 300 pound boilers reusing only journals, wheel centers, and axles plus a few small parts.
The GN purchased 96 FT units by the end of 1945, using them preferentially over the Rocky Mountains, and after the war bought many F3 and F7 units. That displaced the class R engines to the territory between Wenatchee and Spokane, and to Seattle. From Seattle to Skykomish was 1% ruling grade as was the route to Portland, and to Vancouver BC. Most photos of the era show them on the main line to Sky. I doubt that traffic to Vancouver BC was ever heavy enough to need the Rs, but evidently they did get to Portland (or Vancouver WA) on occassion.