Looks like a fairly simple combination of masking (for the elements that are drawn on) scaling and position/rotation adjustments.
Specifically for the sequence at :57-
The outer and inner frame (six sides) have some kind of animated mask on them- if you were to do this in Motion you could go about it a number of ways. Assuming you made the artwork in Photoshop, you would mask each element, and given the simplicity of the shape you could draw a Bezier shape to use as an image mask and then use the Write on behavior to get it to draw the mask.
The wavy line background looks like it is just being masked with a rectangle mask of sorts- you could just animate the Y position of the mask for that.
Do the same with the ropes- just animate the X position of the mask.
The flags at the bottom have animated positions, but are being masked so that they are not visible inside of the frame. Same with 'Director' and the oars.
The wide arrows that fall down behave in the same way, as do the final up/down arrows.
The name looks like it just scales from the first moment of its visibility down to its ending size.