Algorithm. Input: pointers to k unsorted arrays of different lengths. Needed output: k sorted arrays
You can solve the issue by using some pointers. First, run counting sort algorithm on all arrays in $\Theta(n)$ (suppose all of them are in a set). In the meanwhile, when running the algorithm, set a pointer for each number of each array  to the sorted index of that number.

Using this data structure, you will have $k$ sorted arrays at the end.