An infinite-range model of an elastic manifold pulled through a random potential by an applied force $F$ is analyzed focusing on inertial effects. When the inertial parameter, $M$, is small, there is a continuous depinning transition from a small-$F$ static phase to a large-$F$ moving phase. When $M$ is increased to $M_c$, a novel tricritical point occurs. For $M\!>\!M_c$, the depinning transition becomes discontinuous with hysteresis. Yet, the distribution of discrete ``avalanche''-like events as the force is increased in the static phase for $M\!>\!M_c$ has an unusual mixture of first-order-like and critical features.