There are two likely problems. 1) Make sure the midi note in Logic actually starts on or after the beat line. Zoom all the way in on your part in Logic to verify this.
2) For some strange reason that I have not been able to figure out, BB takes dead space at the end of a midi file and moves it to the beginning. See the second post in this string for a description:
Make sure that a drum hit is drawn to go to the end of the midi part in Logic. This will prevent the space shifting.