<p>The multi-dimensional Special Affine Fourier Transformation (SAFT), which generalizes several well-known unitary transformations, has proven to be a valuable tool in signal processing and optics. Previous research has studied the sampling of band-limited signals, but our paper investigates multivariate dynamical sampling in shift-invariant spaces associated with the SAFT. Firstly, we derive various convolution theorems. Then we characterize dynamical sampling in a shift-invariant space through two approaches, the discrete SAFT and the continuous SAFT. In the end, we present two straightforward examples to clarify our main results and demonstrate the superiority of the signal recovery method of the n-dimensional SAFT (nD-SAFT) over the n-dimensional Fourier transform (nD-FT) and the n-dimensional Linear Canonical transform (nD-LCT).</p>