Let $R$ and $S$ be polynomial rings of positive dimensions over a field $k$. Let $I \subseteq R$, $J \subseteq S$ be non-zero homogeneous ideals none of which contains a linear form. Denote by $F$ the fiber product of $I$ and $J$ in $T = R \otimes_k S$. We compute homological invariants of the powers of $F$ using the data of $I$ and $J$. Under the assumption that either $\mathrm{char\,} k = 0$ or $I$ and $J$ are monomial ideals, we provide explicit formulas for the depth and regularity of powers of $F$. In particular, we establish for all $s \geq  2$ the intriguing formula $\mathrm{depth}(T/F^s) = 0$. If moreover each of the ideals $I$ and $J$ is generated in a single degree, we show that for all $s \geq  1$, $\mathrm{reg\,} F^s = \max_{i\in [1, s]}\{\mathrm{reg\,} I^i + s − i, \mathrm{reg\,} J^i + s − i\}$. Finally, we prove that the linearity defect of $F$ is the maximum of the linearity defects of $I$ and $J$, extending previous work of Conca and Römer. The proofs exploit the so-called Betti splittings of powers of a fiber product.