APL (Dyalog Classic), 24 bytes

(t-1)↓(-t←2⍴⎕)∘↑¨,\⍪⍀⍪¨⎕

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the result is a matrix of matrices, though Dyalog's output formatting doesn't make that very obvious

input the matrix (⎕), turn each element into a matrix on its own (⍪¨), take prefix concatenations by row (,\) and by column (⍪⍀), input n (⎕), take the bottom right n-by-n corner from each matrix ((-t←2⍴⎕)∘↑¨), drop the first n-1 rows and columns of matrices ((t-1)↓)