The region $Y$ introduces a delay time $\tau$, the average time between entry and exit. Let me assume that the dynamics in $Y$ is diffusive, with diffusion coefficient $D$. The region $Y$ has area $A$, perimeter $P$, and $W$ is the width of the tunnel. The mean time between collisions with the perimeter of $Y$ is $A/D$, and on average $P/W$ collisions occur between entry and exit, so
$$\tau=C\times\frac{AP}{DW}$$
The shape dependence enters in the coefficient $C$ of order unity, but different shapes with the same area and perimeter will have the same $\tau$ in order of magnitude.
A more precise expression, including the numerical coefficient, can be obtained if the dynamics inside $Y$ is ergodic but ballistic (velocity $v$) rather than diffusive, so the Brownian motion applies only to the tunnel but not inside $Y$. Then
$$\tau=\frac{\pi A}{vW}$$
without any shape dependence.