The resistivity of different films and structures containing fluorinated graphene (FG) flakes and chemical vapor deposition (CVD)-grown graphene of various fluorination degrees under tensile and compressive strains due to bending deformations was studied. Graphene and multilayer graphene films grown by means of the chemical vapor deposition (CVD) method were transferred onto the flexible substrate by laminating and were subjected to fluorination. They demonstrated a weak fluorination degree (F/C lower 20%). Compressive strains led to a strong (one-two orders of magnitude) decrease in the resistivity in both cases, which was most likely connected with the formation of additional conductive paths through fluorinated graphene. Tensile strain up to 3% caused by the bending of both types of CVD-grown FG led to a constant value of the resistivity or to an irreversible increase in the resistivity under repeated strain cycles. FG films created from the suspension of the fluorinated graphene with a fluorination degree of 20–25%, after the exclusion of design details of the used structures, demonstrated a stable resistivity at least up to 2–3% of tensile and compressive strain. The scale of resistance changes ΔR/R0 was found to be in the range of 14–28% with a different sign at the 10% tensile strain (bending radius 1 mm). In the case of the structures with the FG thin film printed on polyvinyl alcohol, a stable bipolar resistive switching was observed up to 6.5% of the tensile strain (bending radius was 2 mm). A further increase in strain (6.5–8%) leads to a decrease in ON/OFF current ratio from 5 down to 2 orders of magnitude. The current ratio decrease is connected with an increase under the tensile strain in distances between conductive agents (graphene islands and traps at the interface with polyvinyl alcohol) and thickness of fluorinated barriers within the active layer. The excellent performance of the crossbar memristor structures under tensile strain shows that the FG films and structures created from suspension are especially promising for flexible electronics.