The following contains SPOILERS for the new live-action Dumbo.
While the new live-action Dumbo may technically be considered a remake, it is, in many ways, a re-imagining. The new film has several entirely new characters and plot threads that never appeared in the original 1941 animated classic, and it also eliminates characters and sequences that were there the first time along.
However, just because the new Dumbo is a very different thing doesn't mean it forgot its roots. There are several great easter eggs and references to the original Dumbo inside the new Tim Burton film. Here are the biggest ones.
Casey Jr.
One element that hasn't changed between the animated original and the brand new Dumbo is the method the circus uses to get from place to place. Casey Jr. is the name of the train. We see the name Casey Jr. on the side of the locomotive as the circus is getting underway at the beginning of the movie. The live action version even has a slightly discomforting face on the front of the locomotive, even more than the animated train does.
Of course, if the train is being used is the same, does it really count as a reference? Maybe not, but what absolutely does is the music. "Casey Junior" is actually one of the songs in the original Dumbo, and one of the better ones, in my opinion. You can hear some pieces of the original music in Danny Elfman's score for the new Dumbo's opening scene. In addition, Danny Devito's character, ringmaster Max Medici, breaks into a chorus of the song at a couple points. Why he came up with a song about his own train, I can't say.
Timothy Q. Mouse
The single biggest change to Dumbo between the two versions of the story concerns the characters that surround the title elephant. In the new movie, it's the family of Holt Ferrier and his two children. However, in the animated classic, Dumbo has no human caretakers. Instead he makes friends with a small mouse who comes to Dumbo's defense when the other elephants of the circus shut out the newborn.
Timothy Q. Mouse spends most of Dumbo in a circus ringmaster's costume. While he might not be a character this time around, we still see a mouse dressed as a ringmaster, as a homage to the character. He's one of the pet mice kept by Holt Ferrier's daughter, Milly.
The Stork
Both the original Dumbo and the new remake start by introducing us to Dumbo's mother, Mrs. Jumbo, before the flying elephant is actually born. Of course, the Disney animated movie isn't going to get into the whole question of where babies come from, so the film introduces us to a stork character who actually delivers little Dumbo to his mother.
While the new Dumbo doesn't dwell on where Dumbo came from exactly, Mrs. Jumbo is already pregnant when we meet her, and we assume she gave birth the old fashioned way. However, before she does, we see her looking out her window, where she spots a stork land just for just a moment before taking flight again and joining its flock. Maybe the stork delivered Dumbo in this movie after all.
The Magic Feather
In the original Dumbo, the elephant and his mouse friend find themselves up a tree after a night of accidental heavy drinking, where they meet a group of crows. While the less said about the crows, the better, they are important for the story because they're the ones who first give Dumbo a feather and claim it has the magic power to make him fly.
In the new Dumbo, there's no belief that the feather is actually magic, but there is a feeling that when Dumbo inhales a feather it grants him his ability to fly. In this movie as well, that turns out to not be true, but the connection is there. Also, the first feather that Dumbo inhales up his trunk which initially launches him into the air is a simple black feather, meant to remind viewers of the crow feather from the original movie.
Pink Elephants on Parade
One of the most famous sequences in the original Dumbo is one that probably gave you nightmares if you first saw it as a child. Dumbo inadvertently drinks some water that has been spiked with booze and the elephant gets drunk. He then blows a large bubble out of his trunk that becomes a group of pink elephants, which then begin to sing and dance and otherwise be terrifying. We have witnessed Dumbo's alcohol-induced hallucination.
In the new Dumbo, we get a similar pink elephant scene. Once again the elephants are created from bubbles. Only this time, the bubbles come from wands being waved by some of V.A. Vandevere's circus performers. Exactly how the bubbles then begin to dance is far from clear. But they do, all while the same music plays. It comes across a bit more as "somebody spiked the popcorn with LSD" than "I really need to quit drinking," but multiple people see the elephants, so it appears that they're quite real.
When I See An Elephant Fly
While Disney has gone ahead and made some live-action remakes like Beauty and the Beast full musicals, Dumbo is not one of those. However, as mentioned, some of those songs are referenced in the new film in other ways. The same crows that give Dumbo his magic feather sing a song in the film, which has a few lines borrowed in the new movie as part of an unexpected cameo.
Famed boxing ring announced Michael Buffer appears as the announcer for V.A. Vadevere's Coliseum, and as part of his build up, he makes references to unlikely things we've all seen fly, such as seeing a "dragon fly" a "horse fly" or a "house fly." These same lines are part of the intro of the song "When I See an Elephant Fly" from the original film. The crows are laughing at the idea of seeing an elephant fly, though they admit they've seen some strange things fly in their time.
In order to turn the brief story of Dumbo into a modern feature film, the story certainly needed work, so it makes sense that a lot of changes and additions were made. However, for the generations that grew up on the previous version, there are a few items to keep your eyes open for.