An electron beam can be injected into the interaction region of a free-electron laser adiabatically or non-adiabatically. Non-adiabatic injection can induce axial velocity fluctuations, therefore adiabatic injection is preferred. For free-electron lasers with axial magnetic fields, initial perpendicular velocities can induce axial velocity fluctuation too. In order to reduce the fluctuation amplitude, a step-like axial magnetic field can be applied for cooling perpendicular temperature of the beam, and proper selection of system parameters is necessary.