How to calculate BIC for multidimensional problem

Sorry for this question, but I am really not sure how to calculate BIC for my situation.

My models are mixtures of normals with different number of components. Variances are equal for all components within the model.

I have 10 datasets of 1000 points each. The datasets are generated by different models in terms of variances, but all models from 10 datasets have the same number of components and the same means of normal components. This is why I can not just mix all 10 datasets in one.

So, I have models $M_1, M_2, ..., M_n$. Only one of them is true for all 10 datasets. Using usual BIC dataset by dataset I can choose, i.e., 8 times $M_1$ and 2 times other models. How can I estimate BIC for 10 datasets simultaneously?

I can calculate the number of parameters: for each dataset it is just

number of estimated means + 1 estimated variance per model + number of estimated proportions

But what should I do with the number of datapoints $n$ in this formula? Should I sum all BICs dataset by dataset, or put all points/sum of log-likelihood into the BIC formula?

$\mathrm{BIC} = {-2 \cdot \ln{\hat L} + k \cdot \ln(n)}. \$

I found BIC calculation here, but I want to understand and to be sure if the procedure from Mclust is valid (for my case) because I am solving several problems separately, it is not a multi-dimensional problem in some sense.