A new method for investigating the dynamics of atomic magnetic moments in current-carrying magnetic point contacts under bias is presented. This combines the non-equilibrium Green's function (NEGF) method for evaluating the current and the charge density with a description of the dynamics of the magnetization in terms of quasistatic thermally-activated transitions between stationary configurations. This method is then implemented in a tight-binding (TB) model with parameters chosen to simulate the main features of the electronic structures of magnetic transition metals. We investigate the domain wall (DW) migration in magnetic monoatomic chains sandwiched between magnetic leads, and for realistic parameters find that collinear arrangement of the magnetic moments of the chain is always favorable. Several stationary magnetic configurations are identified, corresponding to a different number of Bloch walls in the chain and to a different current. The relative stability of these configurations depends on the geometrical details of the junction and on the bias, however we predict transitions between different configurations with activation barriers of the order of a few tens of meV. Since different magnetic configurations are associated to different resistances, this suggests an intrinsic random telegraph noise at microwave frequencies in the I-V characteristics of magnetic atomic point contacts at room temperature. Finally, we investigate whether or not current induced torques are conservative.