Consider the following Normal Dynamic Linear Model (model):
\[\begin{aligned} \bm y_t \mid \bm\theta_t &\sim \MvNormal(\bm F_t^T \bm\theta_t, \bm V_t) \\ \bm \theta_t \mid \bm\theta_{t-1} &\sim \MvNormal(\bm G_t \bm\theta_{t-1}, \bm W_t) \end{aligned}\]for $t\in \bc{1,\dots,T}$, with $\bm\theta_0 \sim \MvNormal(\bm m_0, \bm C_0)$.
Let $\bm D_t = \bc{y_1, \dots, y_T}$. Then at each time $t$, we desire the distribution $\bm\theta_t \mid \bm D_t$, which can be obtained analytically due to all distributions being Normal.
So, at each time point, we have
\[\begin{aligned} \bm y_t \mid \bm\theta_t &\sim \MvNormal(\bm F_t^T \bm\theta_t, \bm V_t) \\ \bm \theta_t \mid \bm\theta_{t-1} &\sim \MvNormal(\bm G_t \bm\theta_{t-1}, \bm W_t) \\ \bm \theta_{t-1} \mid \bm D_{t-1} &\sim \MvNormal(\bm m_{t-1}, \bm C_{t-1}), \end{aligned}\]where, for convenience, $(\bm \theta_{0} \mid \bm D_{0}) = \bm\theta_0$.
We can marginalize over $\bm\theta_{t-1}$ to obtain
\[\begin{aligned} \bm y_t \mid \bm\theta_t &\sim \MvNormal(\bm F_t^T \bm\theta_t, \bm V_t) \\ \bm \theta_t \mid \bm D_{t-1} &\sim \MvNormal( \bm G_t \bm m_{t-1}, ~\bm G_t \bm C_{t-1} \bm G_t ^ T + \bm W_t ) \\ \end{aligned}\]Thus, by applying the content from Bayesian Linear Model , we can obtain $ \bm \theta_t \mid \bm y_t, \bm D_{t-1} = \bm \theta_t \mid \bm D_{t} \sim \MvNormal(\bm m_t, \bm C_t) $, where
\[\begin{aligned} \bm C_t &= \p{ \bm F_t \bm V_t^{-1} \bm F_t^T + \bk{\bm G_t \bm C_{t-1} \bm G_t^T + \bm W_t}^{-1} }^{-1} \\ \bm m_t &= \bm C_t \p{ \bm F_t \bm V_t^{-1} \bm y_t + \bk{\bm G_t \bm C_{t-1} \bm G_t^T + \bm W_t}^{-1} \bm G_t \bm m_{t-1} }. \end{aligned}\]Note that the outer inversion for computing $\bm C_t$ can be done using the Sherman-Morrison-Woodbury formula if the dimensions of $\bm y_t$ is much larger than that of $\bm\theta_t$.