While animation fans are still looking at Disney and other cartoons from the 1930's, they're not always aware of how the cartoons were influenced by what else was happening in the movies at the same time. Animators were going to the movies just like everybody else, and when it came time to create cartoons, they often referred back to films that they'd seen. Steamboat Willie is heavily influenced by Buster Keaton's Steamboat Bill, Jr, both in terms of the setting and also the relationship between Buster/Mickey and the captain of the ship. In Thru the Mirror, the Disney artists were working from the two dominant strains of musicals in the mid-1930's.
After the initial flurry of musicals at the dawn of the talkie era (roughly 1927-1931), musicals fell out of favour. They were revived in two different ways later in the 1930's. At Warner Bros. in 1932, 42nd Street contained musical numbers created by Busby Berkeley. Berkeley wasn't interested in dance so much as he was interested in patterns of motion. He was more interested in moving people, props and the camera around on the screen than he was in presenting fancy footwork. He was also noted for his overhead camera shots. 42nd Street was so successful that Berkeley created the climactic musical numbers for a whole series of Warner musicals as the 1930s progressed.
The other branch of musical was dominated by Fred Astaire and Ginger Rogers starting in 1934 in Flying Down to Rio. Dance in their films was an expression of romantic feelings between the pair or else was played as a novelty. Astaire often danced without Rogers, instead working with props.
Thru the Mirror blends both branches of 1930's musicals, though it leans more heavily on Astaire than it does on Berkeley. Leonard Sebring's scenes (particularly shots 26 through 28) and Ugo D'Orsi's (shots 39 and 40) present masses of cards in patterns of movement in a Berkeley-like fashion. Shot 28 by Sebring is a typical Berkeley overhead shot.
Thru the Mirror, the Astaire influence is heavily felt in Dick Lundy's Mickey scenes. Mickey is surrounded by the accessories of Astaire's costume: the top hat, gloves and cane. Berkeley's dancers might be wearing anything, but Astaire was usually found in formal wear. In particular, there's a very strong influence from the title number in Top Hat (1935) in Mickey's dance animation. Mickey uses a matchstick as a cane in a similar way that Astaire handles his cane, smacking it on the ground for rhythmic effect. Mickey uses it to make the top hat he's dancing on rise and fall. Furthermore, the climax of the "Top Hat" number is Astaire using his cane as a gun, shooting the other dancers. In shot 22, Mickey shoots at the top hat with his cane in a similar way.
Top Hat here if you're interested.