And here's a WxH version. It's interesting what can be found in List::AllUtils...

#!/usr/bin/perl

use strict; # https://theweeklychallenge.org/blog/perl-weekly-challeng
+e-248/#TASK2
use warnings;
use List::AllUtils qw( sum zip_by reductions );
use Data::Dump qw( pp );

sub nslide # size, elements
  {
  my @q = splice @_, 0, shift;
  return reductions { push @q, $b; $a + $b - shift @q } sum(@q), @_;
  }

my ($width, $height) = (2, 2);
for ( [ [1,  2,  3,  4],
        [5,  6,  7,  8],
        [9, 10, 11, 12] ],
      [ [1, 0, 0, 0],
        [0, 1, 0, 0],
        [0, 0, 1, 0],
        [0, 0, 0, 1] ],
    )
    {
    print 'Input: $a = ', pp($_), "\n";
    my @new = zip_by { [ @_ ] } map [ nslide $height, @$_ ],
      zip_by { [ @_ ] } map [ nslide $width, @$_ ], @$_;
    print 'Output: $b = ', pp(\@new), "\n\n";
    }
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