TL;DR: If $\Phi$ and $\Psi$ are quantum channels (unitary or otherwise), then things are "too good to be true".
Proposition. If $\|\Phi-\Psi\|_\diamond\le\varepsilon$ and $m=\max(\|\Phi\|_\diamond, \|\Psi\|_\diamond)\leqslant 1$, then for all $t\in\mathbb{N}$
$$
\|\Phi^{\otimes t}-\Psi^{\otimes t}\|_\diamond\leqslant t\varepsilon.\tag1
$$
Proof. Assume that $\|\Phi^{\otimes t}-\Psi^{\otimes t}\|_\diamond\leqslant t\varepsilon$. Then
$$
\begin{align}
\|\Phi^{\otimes(t+1)}-\Psi^{\otimes(t+1)}\|_\diamond&=\|\Phi\otimes\Phi^{\otimes t}-\Phi\otimes\Psi^{\otimes t}+\Phi\otimes\Psi^{\otimes t}-\Psi\otimes\Psi^{\otimes t}\|_\diamond\tag2\\
&\leqslant\|\Phi\otimes\Phi^{\otimes t}-\Phi\otimes\Psi^{\otimes t}\|_\diamond+\|\Phi\otimes\Psi^{\otimes t}-\Psi\otimes\Psi^{\otimes t}\|_\diamond\tag3\\
&=\|\Phi\|_\diamond\|\Phi^{\otimes t}-\Psi^{\otimes t}\|_\diamond+\|\Phi-\Psi\|_\diamond\|\Psi^{\otimes t}\|_\diamond\tag4\\
&\leqslant mt\varepsilon+m^t\varepsilon\tag5\\
&\leqslant (t+1)\varepsilon\tag6
\end{align}
$$
which completes proof by induction.$\square$
Now, if $\Omega$ is a quantum channel, i.e. a completely positive, trace-preserving, linear (CPTP) map then $\|\Omega\|_\diamond=1$, so we have
Corollary. If $\Phi$ and $\Psi$ are CPTP maps and $\|\Phi-\Psi\|_\diamond\leqslant\varepsilon$, then $\|\Phi^{\otimes t}-\Psi^{\otimes t}\|_\diamond\leqslant t\varepsilon$ for all $t\in\mathbb{N}$.
However, if we drop the assumption that $m\leqslant 1$ then there are counterexamples to $(1)$. Choose any quantum channel $\Psi$ and set $\Phi:=b\Psi$ for $b:=1+\varepsilon>1$. Then $\|\Phi-\Psi\|_\diamond=\varepsilon$, but
$$
\|\Phi^{\otimes t}-\Psi^{\otimes t}\|_\diamond=\|b^t\Psi^{\otimes t}-\Psi^{\otimes t}\|_\diamond=b^t-1>t\epsilon.\tag7
$$
The corollary has important practical implications. Namely, it makes it easy and convenient to obtain fairly tight bounds on error rates of quantum circuits given the knowledge of error rates of component gates. This highly desirable property does not hold for many popular quantities used for describing gate error rates, such as average gate fidelity.