The self-force problem-which asks how self-interaction affects a body's motion-has been poorly studied for spacetime dimensions d not equal 4. We remedy this for all d >= 3 by nonperturbatively constructing momenta such that forces and torques acting on extended, self-interacting electromagnetic charges have the same functional forms as their test body counterparts. The electromagnetic field which appears in the resulting laws of motion is not however the physical one, but a certain effective surrogate which we derive. For even d >= 4, explicit momenta are identified such that this surrogate field satisfies the source-free Maxwell equations; laws of motion in these cases can be obtained similarly to those in the well-known four-dimensional Detweiler-Whiting prescription. For odd d, no analog of the Detweiler-Whiting prescription exists. Nevertheless, we derive its replacement. These general results are used to obtain explicit point-particle self-forces and self-torques in Minkowski spacetimes with various dimensions. Among various characteristics of the resulting equations, perhaps the most arresting is that an initially stationary charge which is briefly kicked in 2 + 1 dimensions asymptotically returns to rest.