The answer will, at least in part, depend on the massage technique you are using for the myofascial fibers.
The principal objective of friction, or “stripping” massage (gliding the finger(s), hand or tool parallel to the fibers), is to empty the venous and lymphatic channels, which encourages better circulation at the deepest layers of the tissue.
Cross-fiber work (Active Release Technique or ART, “deep tissue”, etc.) mainly addresses the breakup of adhesive connective tissue.
Reference: “The Muscular Force Transmission System: Role of the Intramuscular Connective Tissue”, by Andrea Turrina, PT, Miguel Antonio Martínez-González, PT, PhD, and Carla Stecco, MD. Published online on June 7, 2012