In addition to other answers.
But what about tempo? I've indicated a metric tempo at the start of the piece—but in 4/4, each quarter note gets one beat, and in 9/8, each eighth note gets a beat. So at the same tempo, you're suddenly going twice as fast, right?
I guess you are used to define tempo in "beats per minute", but in sheet music it is typically specified as number of notes of given value – e.g. quarter notes – per minute. Tempo specified this way does not change.
Also, if 9/8 is in fact a compound meter of 4/4 + 1/8, possibly the beat base doesn't even change, except for the time when it needs to shift by an eight note.