Continuity mistake: The father tears the children's advert for a nanny up into 8 pieces, but when it comes out of the chimney it's in far more bits.
Factual error: The Robin that comes along in the movie is an American Robin, not a British Robin.
Continuity mistake: At the bank where the elder Mr. Dawes appears from the back room, he makes it as far as the edge of the step in that shot. In the next shot, he's back by the door walking again to the edge of the step.
Revealing mistake: When all four actors are up on the rooftops, look closely at them when they all step onto the smoke platform that brings them down. All four actors are doubles. (01:46:50)
Revealing mistake: An early scene with the constable and the children, Jane silently mouths along with one of Michael's lines.
Continuity mistake: When Uncle Albert, Bert, and the kids fall down from hovering on the ceiling, they're about to land on the chairs. But when the camera view changes, they land on the floor.
Other mistake: In the ceiling tea party, Mary Poppins offers Bert some sugar, and when he responds that he would not like any, his Cockney accent is gone.
Revealing mistake: When Mary Poppins takes the children on an outing to the park, look at the front steps when the front door opens. They are not steps but flat ground painted to look like steps. (00:36:52)
Revealing mistake: In the scene where Mary Poppins is removing the large plant from her carpet bag, Michael inspects the underside of the table, where the table leg casts a shadow across his face. When Mary crosses in front of him (still under the table) she does NOT cast a shadow, although she should.
Continuity mistake: When the kids meet Bert as a pavement artist, he draws a road with a single-arched bridge on his drawing. When they jump into the drawing, the bridge is double arched.
Continuity mistake: At the end of the film, Mary Poppins flies away from the Banks's house (supposedly within a short distance of the City of London), however, she is then shown flying towards the City on a route that would take her over the Banks's house on the way, had she not already flown over it.
Continuity mistake: In the beginning, when Mr. Banks is calling the police and talking to his wife at the same time, the phone cord changes position between shots.
Revealing mistake: At the very beginning of the movie, when Mary Poppins is sitting on the cloud, you can see the string attached to her bag just as it almost falls through the cloud.
Continuity mistake: When Mr. Banks throws the kids' letter pieces to the fireplace, their position changes between shots.
Continuity mistake: When Mary reads the measuring tape after measuring herself the message reads up and down (as shown to the audience), but then the camera shows Mary reading it side to side.
Revealing mistake: There is a certain point in the song "Step In Time" (I believe it is when we see a couple of men switching their feet on the top of a chimney) where against the sky you can see where a wall and ceiling meet. This is because the scene was shot indoors.
Revealing mistake: During the close-up shots of the ceiling tea party, Bert and Uncle Albert have shadows on the wall directly behind them. However in most of the wider shots, (which have obviously been shot against a blue or green screen) the shadows are now gone.
Continuity mistake: When Mr. Banks begins singing "Let's Go Fly A Kite," Ellen the maid puts her left hand up to her cheek in surprise. In the very next shot, her hands are clasped in front of her.
Revealing mistake: In the tea party on the ceiling sequence, 2 shots after Uncle Albert tells his long underwear joke, Jane does a backflip in the air. If you look closely you can see the imprints that wires that she was hanging from made on her dress. (01:15:55)
Continuity mistake: The "Votes for women" band over the old nanny's shoulder swaps positions when the song about suffrage is about to end.
Answer: Most likely a variation on the "inexhaustible bottle" magic trick which has existed in various forms for hundreds of years - it relies on different chambers holding different liquids, and air holes in the bottle being covered or not change the pressure and allow the desired liquid to flow out. As the scene is one continuous shot with the entire bottle visible, this seems most likely - the air holes could be on the side away from the camera, with Julie Andrews repositioning her thumb for each colour. More info here: https://en.wikipedia.org/wiki/Inexhaustible_bottle.