Convert n same-size, d-dimensional numpy arrays to a dataframe with d+n columns

Question

I recently asked this question, about converting n 2-dimensional arrays to a dataframe with 2+n columns. The solution I got works perfectly well, but can not easily be generalized to higher dimensions.

In the more general case, the code would take as input a list of n same-size, d-dimensional numpy arrays, and d lists whose lengths correspond to the sizes of the arrays in the corresponding dimensions. The output is a pandas dataframe where each row corresponds to one position in the d-dimensional array. The first d comlumns contain the coordinates from the lists, and the next n columns contain the corresponding values from the n arrays. The linked question contains an example in 2D.

How would I go about generalizing the code to the d-dimensional case? I feel like this problem is related to the Pandas melt function, but I'm not sure how to make it work.