  Let $\mathbb{X}=(\mathbb{X}_t)_{t\geq 0}$ be the subdiffusive process
defined, for any $t\geq 0$, by $ \mathbb{X}_t = X_{\ell_t}$ where
$X=(X_t)_{t\geq 0}$ is a L\'evy process and $\ell_t=\inf \{s>0;\:
\mathcal{K}_s>t \}$ with $\mathcal{K}=(\mathcal{K}_t)_{t\geq 0}$ a subordinator
independent of $X$. We start by developing a composite Wiener-Hopf
factorization to characterize the law of the pair
$(\mathbb{T}_a^{(\mathcal{b})}, (\mathbb{X} -
\mathcal{b})_{\mathbb{T}_a^{(\mathcal{b})}})$ where
  \begin{equation*} \mathbb{T}_a^{(\mathcal{b})} = \inf \{t>0;\: \mathbb{X}_t >
a+ \mathcal{b}_t \} \end{equation*} with $a \in \mathbb{R}$ and
$\mathcal{b}=(\mathcal{b}_t)_{t\geq 0}$ a (possibly degenerate) subordinator
independent of $X$ and $\mathcal{K}$. We proceed by providing a detailed
analysis of the cases where either $\mathcal{K}$ is a stable subordinator or
$X$ is spectrally negative. Our proofs hinge on a variety of techniques
including excursion theory, change of measure, asymptotic analysis and on
establishing a link between subdiffusive processes and a subclass of
semi-regenerative processes. In particular, we show that the variable
$\mathbb{T}_a^{(\mathcal{b})}$ has the same law as the first passage time of a
semi-regenerative process of L\'evy type, a terminology that we introduce to
mean that this process satisfies the Markov property of L\'evy processes for
stopping times whose graph is included in the associated regeneration set.
