Understanding strain gradient phenomena is of paramount importance in diverse areas of condensed matter physics. This effect is responsible for flexoelectricity in dielectric materials, and it plays a crucial role in the mechanical behavior of nanoscale-sized specimens. In magnetoelectric composites, which comprise piezoelectric or ferroelectric (FE) materials coupled to magnetostrictive (MS) phases, the strain gradient can add to any uniform strain that is present to boost the strength of the coupling. Hence, it could be advantageous to develop new types of functionally graded multiferroic composites (for information technologies) or magnetic-field-driven flexoelectric/magnetostrictive platforms for wireless neurons/muscle cell stimulation (in biomedicine). In MS or FE materials with non-fully constrained geometries (e.g., cantilevers, porous layers, or vertically aligned patterned films), strain gradients can be generated by applying a magnetic field (to MS phases) or an electric field (to, e.g., FE phases). While multiferroic composites operating using uniform strains have been extensively investigated in the past, examples of new nanoengineering strategies to achieve strain-gradient-mediated magnetoelectric effects that could ultimately lead to high flexomagnetoelectric effects are discussed in this Perspective.