Among efforts to improve techniques for the chemical vapor deposition of large-area and high-quality graphene films on transition metal substrates, being able to reliably transfer these atomistic membranes onto the desired substrate is a critical step for various practical uses, such as graphene-based electronic and photonic devices. However, the most used approach, the wet etching transfer process based on the complete etching of metal substrates, remains a great challenge. This is mainly due to the inevitable damage to the graphene, unintentional contamination of the graphene layer, and increased production cost and time. Here, we report the systematic study of an H2 bubbling-assisted transfer technique for graphene films grown on Cu foils, which is nondestructive not only to the graphene film but also to the Cu substrate. Also, we demonstrate the origin of the graphene film tearing phenomenon induced by this H2 bubbling-assisted transfer process. This study reveals that inherent features are produced by rolling Cu foil, which cause a saw-like corrugation in the poly(methyl methacrylate) (PMMA)/graphene stack when it is transferred onto the target substrate after the Cu foil is dissolved. During the PMMA removal stage, the graphene tearing mainly appears at the apexes of the corrugated PMMA/graphene stack, due to weak adhesion to the target substrate. To address this, we have developed a modified heat-press-assisted transfer technique that has much better control of both tearing and the formation of residues in the transferred graphene films.