Product of a Multinomial done using Multinomial Theorem?

Question

I was looking into the multinomial theorem to try to understand more about summing a sequence over a composition of a number with non-negative elements (a weak composition). In particular, I am curious about multiplying two of the same multinomial sums and combine the summations somehow into a single term such as the Cauchy product. $$ (x_1+x_2+\ldots+x_m)^a (x_1+x_2+\ldots+x_m)^b = \left(\sum_{\pi_m \in C_a} {a \choose \pi_a} \prod_{v=1}^m x_m^{\lambda_m}\right)\left(\sum_{\pi_m \in C_b} {b \choose \pi_a} \prod_{v=1}^m x_m^{\lambda_m}\right) $$ I intuitively do not understand how multiplying a sum over a weak composition would work. I realized this can easily be solved using algebra or other trivial methods, but I am looking to understand how this method would work to create the solution.