In their seminal 1983 paper, M. Maxey and J. Riley introduced an equation for the motion of a sphere through a fluid. Since this equation features the Basset history integral, the popularity of this equation has broadened the use of a certain form of fractional differential equation to study inertial particle motion. In this paper, we give a comprehensive theoretical analysis of the Maxey-Riley equation. In particular, we build on previous local in time existence and uniqueness results to prove that solutions of the Maxey-Riley equation are global in time. In doing so, we also prove that the notion of a maximal solution extends to this equation. We furthermore prove conditions under which solutions are differentiable at the initial time. By considering the derivative of the solution with respect to the initial conditions, we perform a sensitivity analysis and demonstrate that two inertial trajectories can not meet, as well as provide a control on the growth of the distance between a pair of inertial particles. The properties we prove here for the Maxey-Riley equations are also possessed, mutatis mutandis, by a broader class of fractional differential equations of a similar form.