  Subdiffusion equation and molecule survival equation, both with Caputo
fractional time derivatives with respect to another functions $g_1$ and $g_2$,
respectively, are used to describe diffusion of a molecule that can disappear
at any time with a constant probability. The process can be interpreted as
``ordinary'' subdiffusion and ``ordinary'' molecule survival process in which
timescales are changed by the functions $g_1$ and $g_2$. We derive the
first-passage time distribution for the process. The mutual influence of
subdiffusion and molecule vanishing processes can be included in the model when
the functions $g_1$ and $g_2$ are related to each other. As an example, we
consider the processes in which subdiffusion and molecule survival are highly
related, which corresponds to the case of $g_1\equiv g_2$.
