Don't need a .blend file for this one.
If I open an .avi in an image node in the compositer, specify the number of frames (e.g. 30) and check "Cycle" everything works fine. It repeats the section properly.

If I open an .avi in an image node in the compositer, specify the number of frames (e.g. 30) and set an "Offset" value, without "Cycle," everything works fine. It starts the "Movie" at the correct "Offset" frame value properly.

However, if you set both "Cycle" and an "Offset" frames value, it doesn't cycle from the offset point properly. It repeats the correct number of frames selected from the start of the file, frame 1. It should cycle starting from the "Offset" the number of frames selected.