We extend a theorem of the second author on the $L^q$-dimensions of dynamically driven self-similar measures from the real line to arbitrary dimension. Our approach provides a novel, simpler proof even in the one-dimensional case. As consequences, we show that, under mild separation conditions, the $L^q$-dimensions of homogeneous self-similar measures in $\mathbb{R}^d$ take the expected values, and we derive higher rank slicing theorems in the spirit of Furstenberg's slicing conjecture.