Continuity mistake: When the Turtles are climbing the wall to get into the garbage-chute that leads to the prison, you can see Donatello's bandana tail hanging down towards the wall just as if they were crawling on the floor. (01:09:05)
Continuity mistake: When Raphael meets Yoshi and talks to him about being a kid and having fun, if you look at Raph's wrists you'll notice that he is not wearing any wrist bands as he usually does as with the rest of the turtles. However, when Raph puts his hand on Yoshi's shoulder there is a close-up of Yoshi's head and shoulders. Look at Raph's wrist in the close-up and you'll see that a wrist band has appeared. When the shot changes back, his wrists are bare again. (00:56:50)
Continuity mistake: When the turtles go sliding down the ramp down into the sewer, watch the inside of the ramp before Raph comes out and you can tell where they stooped recording and restarted it. The shadows seem to change places. This also happens right before Leo comes out too. (00:34:00)
Audio problem: When Leonardo stops Daimyo, the words "Hold it, Bucko" are heard, but Leonardo is not seen to be speaking at all. (01:15:30)
Audio problem: When Walker confronts the Turtles in the Daimyo's courtyard, he says "My God, who's your tailor?" One of the Turtles says "We're naked." but none of the Turtles' mouths move. (01:17:45)
Continuity mistake: Whit and April are fighting by the spring, and just before it cuts to Mitsu preparing an arrow, Elias Koteas is pounced by a guy wearing colorful rags in black, orange and yellow patchwork, a blue and white pattern bandana and black and white striped legwarmers. Behind Mitsu, meanwhile...a guy with the same horrid fashion taste is charging. What are the odds. (00:39:35)
Other mistake: After Leonardo revives Yoshi, he picks him up, hugs him and touches heads with each other as they hug but Yoshi pretty much moved his head immediately and Leonardo continued sitting there as if their heads were still touching. It looks like the actor didn't realise the child moved away and they never reshot it. (00:48:15)
Continuity mistake: When Michelangelo's rolled over by Mitsu after being knocked off his horse, his samurai mask is under his chin. A shot later, after the camera's on Mitsu, it's suddenly gone.
Continuity mistake: When Mitsu removes Raph's helmet, she grabs the lower edge of it. Shot changes and her hand is positioned higher.
Continuity mistake: April is moving the iron sewer door but from the opposite angle the door is lying still on the floor despite her pants of effort and despite the fact that a shot later the door has been moved.
Continuity mistake: When Michelangelo is about to be lifted on the cart Mitsu raises her hand and commands something to the soldiers. A shot later her hand is lowered.
Continuity mistake: When Michelangelo is about be knocked down by Mitsu, the galloping horse arrives at where the camera is and comes to a stop. When the shot changes he is galloping again.
Continuity mistake: When Michelangelo drops the sceptre it lies perpendicular to some green leaves. When Mitsu picks it up the leaves are on the left.
Continuity mistake: Mitsu picks up the arch that's lying against the tree, but it wasn't there when she knocked down Michelangelo.
Continuity mistake: When unconscious Michelangelo is lifted up and put on a cart, a man suddenly appears on the left.
Revealing mistake: When Mitsu knocks down Michelangelo, he falls on the floor and bounces, revealing the safety matt below.
Continuity mistake: When Michelangelo's horse is about to stop galloping, the wide shot shows his right hand lowered and his body bent downwards, but in the close-up his hand is raised. This swaps back and forth. Seemingly, the wide shots were of a puppet of some sort used for security reasons.
Continuity mistake: When ninja-Michelangelo is on the horse, the sceptre swaps hands between shots. (00:25:05)
Continuity mistake: When the soldiers smash April's Walkman, pieces get scattered all over the place. In the close-up they're very close together.
Answer: Michelangelo wants to stay behind for the same reasons as Raphael: because in that time and place, they are well respected, even revered as heroes. They can live openly and not be banished to sewers or subway stations.
LorgSkyegon
Mikey didn't want to stay behind because of Yoshi but because of Mitsu. Raphael even points out that she's the reason he doesn't want to return. When Mikey says they should stay, Raph responds, "Now, forget about Mitsu and give me the scepter."