How to multiplicate two block matrices with following conditions?

I have a block diagonal matrix P with size 500*500 like this

P=[A 0 0 
   0 A 0 
   0 0 A...
   ...     ];

each element A on the diagonal is a (m*n) matrix and there are T number of these matrices. (i.e. There are T elements on the diagonal and T is a large number like 500) I want to multiply each diagonal element(which is a matrix), with one of the elements of another block matrix V that is like this:

   V=[B1;B2;...,BT] 
   %there are again so many Bi; (T=500);
   %Each Bi is a (n*1) vector

Each A on the diagonal should be multiplied to the corresponding B.

In other words we want to calculate a block by block multiplication. One block of matrix P is multiplied by the corresponding block of V. Although A is (m*n) and Bi is (n*1), but when we do P*V, it gives this Error: "Inner matrix dimensions must agree."

[James Tursa]

"... I want to multiply each diagonal element(which is a matrix), with each one of the elements of another block matrix ... "

"... There exists one A for each B ..."

So, which is it? Multiply each A by each B, or multiply only one A for each B? Can you write down in pseudo-code what the result would look like?

[sn at]

 Multiply each A by each B, But there are 500 of these pairs A and Bi.
the answer will almost be like this:
[y1;y2;...;y500] in which yi=A*Bi.
A is (m*n) and each Bi is n*1;
There will be 500 matrices yi which are (m*1) each.

Answer 1

[sn at]

I found a file named block diagonal multiplication by David Holdaway in File Exchange. This function is Used for multiplying large block diagonal matrices with matrices / vectors. I think I will need this.