  We establish an equivalence between two different quantum quench problems,
the joining local quantum quench and the M\"obius quench, in the context of
$(1+1)$-dimensional conformal field theory (CFT). Here, in the former, two
initially decoupled systems (CFTs) on finite intervals are joined at $t=0$. In
the latter, we consider the system that is initially prepared in the ground
state of the regular homogeneous Hamiltonian on a finite interval and, after
$t=0$, let it time-evolve by the so-called M\"obius Hamiltonian that is
spatially inhomogeneous. The equivalence allows us to relate the time-dependent
physical observables in one of these problems to those in the other. As an
application of the equivalence, we construct a holographic dual of the M\"obius
quench from that of the local quantum quench. The holographic geometry involves
an end-of-the-world brane whose profile exhibits non-trivial dynamics.
