Continuity mistake: Leonardo ducks the cannon shot tucking his head inside the shell with the bandana going in front of his shoulder, but he emerges with the bandana on both sides of the neck later. (01:19:50)
Continuity mistake: The goons with guns arrive at the castle, "There goes the neighbourhood", quips one of the heroes. In one shot Raphael sidesteps towards Michelangelo to reach his arm, in the reverse it's Mike and Don who are restraining Raph from charging. Same mistake happens a few seconds later, when Walker delivers his "Who's your tailor" line and Don is restraining Raph, or not. (01:17:35)
Continuity mistake: In the fight alongside Mitsu, Donatello beats up some soldiers by the table with the scrolls ("Bases loaded, home run"). The scroll is placed in a different, more orderly way than before (when it was on top of another parchment). It wouldn't tidy itself up during the fight. (01:13:40)
Continuity mistake: After learning about their supposed ancestors, the Turtles begin fighting Norinaga's guards. Michelangelo is the first one in their way and is at a standoff with one of them who has his sword raised, and still the following shot shows exactly Mike facing some enemies that run at him. (01:13:10)
Continuity mistake: Raphael points out that Yoshi is no 'senior citizen'. He adds "Come ooooonnn..." and in the passage from one angle to another Raph's bandana moved from the front to the back of his shoulder. (00:56:50)
Continuity mistake: Leonardo picks Yoshi up from the ground after reviving him; his bandana goes down his left shoulder, but in close-up goes down both sides of his neck. (00:48:30)
Continuity mistake: In Niles' POV, during their first confrontation in the middle of the village raid Walker on his horse is in the shade, facing Michelangelo in sunlight. It's the opposite in their respective close-ups. (00:44:15)
Continuity mistake: April screams and squirms to free herself from the arrow that pins her sleeve to the tree. Her hand is in a different position in each shot. (00:38:30)
Continuity mistake: The Turtles are fighting the rebels by the springs. The first time we see Leonardo fight some of them off, his first action is hitting (with the side of the sword, we assume, or he'd slice it) the head of a guy, but the 'hit' is really a miss, since no contact is even remotely made. (00:38:20)
Continuity mistake: When Michelangelo loses consciousness and his helmet, the helmet rolls on the ground above his head, but when the rebels are approaching to pick him up and haul him on the cart the helmet is by his shoulder. (00:26:10)
Continuity mistake: When Michelangelo shouts "Woah! Good horsey! Good horsey!" the scepter goes from his left hand to his right between shots. (00:25:35)
Continuity mistake: Walker witnesses the music playing and the guards wrecking the demonic machine; he constantly has his left hand held up in the air in the close-ups, but in the other angle he shifted that hand to lean against the door. (00:15:45)
Continuity mistake: April's walkman is very conveniently prone to play randomly. She puts it down on the floor, but it is, together with the headphones, positioned differently practically at every shot in particular when the guards start bashing it and everything is closer together in close-up than in wider angles. (00:15:30)
Continuity mistake: In the sequence set in present time when the Turtles interact with Kenshin, once he faints seeing the giant Japanese-speaking rat the shopping bags on the table changed position - the blue bag the wind knocked on the floor is back, and the red bag is up and not horizontal. (00:12:00)
Continuity mistake: Leonardo shouts "He's a man!" when Kenshin appears in the Turtles' hideout; he is holding in the foreground with the right hand his book about katanas, but in the next shot it's on the table, under his left hand. (00:11:35)
Continuity mistake: April materializes in the same spot as Kenshin; her headset is placed away from the collar of the kimono. She is still in the same position when she spots the priest, but the collar is underneath the headphones now. (00:11:50)
Continuity mistake: Kenshin throws a temper tantrum in a room with several statues. The doors open and a strong wind blows; the position of the scrolls on the table changes between shots; they are first closed by the edge of the table, then unfurl from a position much further behind. (00:10:10)
Continuity mistake: Walker's henchmen dump on the court Lord Norinaga's sentinels and walk away. They are already shown posing with evil smirks and all that in the next shot while Walker is doing his threatening speech. (00:08:55)
Continuity mistake: We first see Lord Norinaga in a close-up looking at his son's arrival. His helmet is tied under his chin, but when he walks in the helmet's cord is split above and below the chin. (00:07:40)
Continuity mistake: During the introduction of the turtles, Michaelangelo's bandana goes from the back of the shoulder to the front and then both ends behind the shoulders between shots. (00:04:30)
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."