In this paper, we study the problem of pathfinding on traversal-dependent graphs, i.e., graphs whose edges change depending on the previously visited vertices. In particular, we study \emph{self-deleting graphs}, introduced by Carmesin et al. (Sarah Carmesin, David Woller, David Parker, Miroslav Kulich, and Masoumeh Mansouri. The Hamiltonian cycle and travelling salesperson problems with traversal-dependent edge deletion. J. Comput. Sci.), which consist of a graph $G=(V, E)$ and a function $f\colon V\rightarrow 2^E$, where $f(v)$ is the set of edges that will be deleted after visiting the vertex $v$. In the \textsc{(Shortest) Self-Deleting $s$-$t$-path} problem we are given a self-deleting graph and its vertices $s$ and $t$, and we are asked to find a (shortest) path from $s$ to $t$, such that it does not traverse an edge in $f(v)$ after visiting $v$ for any vertex $v$. We prove that \textsc{Self-Deleting $s$-$t$-path} is NP-hard even if the given graph is outerplanar, bipartite, has maximum degree $3$, bandwidth $2$ and $|f(v)|\leq 1$ for each vertex $v$. We show that \textsc{Shortest Self-Deleting $s$-$t$-path} is W[1]-complete parameterized by the length of the sought path and that \textsc{Self-Deleting $s$-$t$-path} is \W{1}-complete parameterized by the vertex cover number, feedback vertex set number and treedepth. We also show that the problem becomes FPT when we parameterize by the maximum size of $f(v)$ and several structural parameters. Lastly, we show that the problem does not admit a polynomial kernel even for parameterization by the vertex cover number and the maximum size of $f(v)$ combined already on 2-outerplanar graphs.