In this paper, we consider the stabilization of an axially moving Kirchhoff-type beam with rotational inertia under controls of force and torque at one boundary. The proposed negative feedbacks of the transverse velocity and angular velocity applied at the control end cover a large class of nonlinear feedback functions. The well-posedness of the resulting closed-loop system is established by means of the nonlinear semigroup theory, where the solution is shown to be depending continuously on the initial value. The asymptotic stability of the closed-loop system is guaranteed by resolving a dissipative ordinary differential equation. The decay rates of the vibration for some special nonlinear feedback functions can be estimated by the dissipative ordinary differential equation provided that growth restrictions on these nonlinear feedbacks near the origin are required. Three types of examples including exponential, polynomial and polynomial-logarithmic decay forms are deduced, and the numerical simulations are presented to verify the proposed control approach.