This study investigates the stability to tilts (misalignments) of Fourier-based multi-pass amplifiers, i.e., amplifiers where a Fourier transform is used to transport the beam from pass to pass. Here, the stability properties of these amplifiers to misalignments (tilts) of their optical components has been investigated. For this purpose, a method to quantify the sensitivity to tilts based on the amplifier small-signal gain has been elaborated and compared with measurements. To improve on the tilt stability by more than an order of magnitude a simple auto-alignment system has been proposed and tested. This study, combined with other investigations devoted to the stability of the output beam to variations of aperture and thermal lens effects of the active medium, qualifies the Fourier-based amplifier for the high-energy and the high-power sector.