Given a smooth globally hyperbolic (3+1)-dimensional spacetime (M,g) satisfying the Einstein vacuum equations (possibly with cosmological constant) and an inextendible timelike geodesic C, we construct, on any compact subset of M, solutions g_\epsilon of the Einstein equations which describe a mass ϵ Kerr black hole traveling along C. More precisely, away from C one has g_\epsilon\to g as \epsilon\to 0, while the \epsilon^{-1}-rescaling of g_\epsilon around every point of C tends to a fixed subextremal Kerr metric. Our result applies on all spacetimes with noncompact Cauchy hypersurfaces, and also on spacetimes which do not admit nontrivial Killing vector fields in a neighborhood of a point on the geodesic. As an application, we construct spacetimes which model the merger of a very light subextremal Kerr black hole with a slowly rotating unit mass Kerr(-de Sitter) black hole, followed by the relaxation of the resulting black hole to its final Kerr(-de Sitter) state. In Part I, we constructed approximate solutions g_{0,\epsilon} of the gluing problem which satisfy the Einstein equations only modulo \mathcal{O}(\epsilon^\infty) errors. Part II introduces a framework for obtaining uniform control of solutions of linear wave equations on ϵ-independent precompact subsets of the original spacetime (M,g). In this final part, we show how to correct g_{0,\epsilon} to a true solution g_\epsilon by adding a metric perturbation of size \mathcal{O}(\epsilon^\infty) which solves a carefully chosen gauge-fixed version of the Einstein vacuum equations. The main novel ingredient is the proof of suitable mapping properties for the linearized gauge-fixed Einstein equations on subextremal Kerr spacetimes.
