  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 $\text{char}~ k=0$ 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 $s\ge 2$ the
intriguing formula $\text{depth}(T/F^s)=0$. If moreover each of the ideals $I$
and $J$ is generated in a single degree, we show that for all $s\ge 1$,
$\text{reg}~ F^s=\max_{i\in [1,s]}\{\text{reg}~ I^i+s-i,\text{reg}~ J^i+s-i\}$.
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 R\"omer.
The proofs exploit the so-called Betti splittings of powers of a fiber product.
