Stagger, stack, sum

R, 88 81 bytes

function(A,N,P,o=0*diag(P*(N-1)+1)){for(i in 1:P)o[x,x]=o[x<-1:N+i-1,x]+A[[i]];o}

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Takes a list of matrices, A, N, and P.

Builds the requisite matrix of zeros o and adds elementwise the contents of A to the appropriate submatrices in o.

JavaScript (ES6), 102 bytes

Takes input as (n,p,a).

(n,p,a)=>[...Array(--n*p+1)].map((_,y,r)=>r.map((_,x)=>a.map((a,i)=>s+=(a[y-i*n]||0)[x-i*n]|0,s=0)|s))

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How?

Redimensioning matrices filled with a default constant (\$0\$ in that case) is neither very easy nor very short in JS, so we just build a square matrix with the correct width \$w\$ right away:

$$w=(n-1)\times p+1$$

For each cell at \$(x,y)\$, we compute:

$$s_{x,y}=\sum_{i=0}^{p-1}{a_i(x-i\times (n-1),y-i\times (n-1))}$$

where undefined cells are replaced with zeros.

Jelly, 15 12 bytes

⁹ṖŻ€ƒZƲ⁺+µ@/

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How it works

⁹ṖŻ€ƒZƲ⁺+µ@/  Main link. Argument: A (array of matrices)

         µ    Begin a monadic chain.
          @/  Reduce A by the previous chain, with swapped arguments.
                Dyadic chain. Arguments: M, N (matrices)
      Ʋ           Combine the links to the left into a monadic chain with arg. M.
⁹                 Set the return value to N.
 Ṗ                Pop; remove its last row.
     Z            Zip; yield M transposed.
    ƒ             Fold popped N, using the link to the left as folding function and
                  transposed M as initial value.
  Ż€                Prepend a zero to each row of the left argument.
                    The right argument is ignored.
       ⁺        Duplicate the chain created by Ʋ.
        +       Add M to the result.