  The standard setup for single-file diffusion is diffusing particles in one
dimension which cannot overtake each other, where the dynamics of a tracer
(tagged) particle is of main interest. In this article we generalise this
system and investigate first-passage properties of a tracer particle when
flanked by crowder particles which may, besides diffuse, unbind (rebind) from
(to) the one-dimensional lattice with rates $k_{\rm off}$ ($k_{\rm on}$). The
tracer particle is restricted to diffuse with rate $k_D$ on the lattice. Such a
model is relevant for the understanding of gene regulation where regulatory
proteins are searching for specific binding sites ona crowded DNA. We quantify
the first-passage time distribution, $f(t)$ ($t$ is time), numerically using
the Gillespie algorithm, and estimate it analytically. In terms of our key
parameter, the unbinding rate $k_{\rm off}$, we study the bridging of two known
regimes: (i) when unbinding is frequent the particles may effectively pass each
other and we recover the standard single particle result $f(t)\sim t^{-3/2}$
with a renormalized diffusion constant, (ii) when unbinding is rare we recover
well-known single-file diffusion result $f(t)\sim t^{-7/4}$. The intermediate
cases display rich dynamics, with the characteristic $f(t)$-peak and the
long-time power-law slope both being sensitive to $k_{\rm off}$.
