Dumbo performs at the big tent and flies out breaking the front doors with glass flying. A few seconds later people are leaving the same show out the same doors which are not damaged.
When Dumbo escapes from the circus with Milly's help, he flies over the Brooklyn Bridge and in the background the Empire State and Chrysler Buildings are shown. The Empire State started construction in 1930 and the Chrysler Building started construction in 1928 yet the movie is set in 1919 and early 1920s.
The first time Dumbo flies inside the big tent, some of the clowns high-five each other. That gesture did not exist in 1919.
When Holt arrives at the train station he is wearing three medals, a Croix de Guerre, an Army Distinguished Service Cross, and a World War I Victory Medal. Although the Victory Medal was already earned by 1919, the Army did not issue the medals until June of 1920. Also the Victory Medal would probably have a Citation Star since he had earned the DSC.