We give an existence proof for variational solutions $u$ associated to the total variation flow. Here, the functions being considered are defined on a metric measure space $(\mathcal{ X}, d, \mu)$ satisfying a doubling condition and supporting a Poincar\'e inequality. For such parabolic minimizers that coincide with a time-independent Cauchy-Dirichlet datum $u_0$ on the parabolic boundary of a space-time-cylinder $\Omega \times (0, T)$ with $\Omega \subset \mathcal{ X}$ an open set and $T > 0$, we prove existence in the weak parabolic function space $L^1_w(0, T; \mathrm{ BV}(\Omega))$. In this paper, we generalize results from a previous work by B\"ogelein, Duzaar and Marcellini by introducing a more abstract notion for $\mathrm{ BV}$-valued parabolic function spaces. We argue completely on a variational level.
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