Calculating covariance matrix for MVDR beamforming
MVDR is a narrowband beamformer. For broadband signals it is usually applied for each frequency bin. That means that $\mathbf{R}_{xx}$ is frequency dependent. In other words, for each time you should have $M$ matrices, each one is $3\times 3$.

Now, since you usually cannot compute $\mathbf{R}_{xx}$ exactly, you perform covariance estimation $\tilde{\mathbf{R}}_{xx}$. For more information you can look here for example. In short, as you mention you perform averaging of the covariance matrix in order to improve the estimation. In order to adapt you estimation to time-varying statistics, usually exponential averaging is performed, i.e.

$$
\tilde{\mathbf{R}}_{xx}[n] = \alpha\tilde{\mathbf{R}}_{xx}[n-1] + (1-\alpha)\mathbf{x}[n]\mathbf{x}^H[n]
$$

for some $0&lt;\alpha&lt;1$ (usually closer to 1).