Column-wise summation of overlapping slices
J, 11, 9 8 bytes

-1 byte thanks to miles!

[:+//.]\


How it works?

The left argument is s, the right one - L

]\ - splits L into sublists with length s

/. - extracts the oblique diagonals (anti-diagonals)

+/ - adds them up

[: - makes a fork from the above verbs

Here's an example J session for the first test case:

   a =. 1 2 3 4 5 6 7 8 9

   ] 3 ]\ a 
1 2 3
2 3 4
3 4 5
4 5 6
5 6 7
6 7 8
7 8 9

   ] &lt;/. 3 ]\ a 
┌─┬───┬─────┬─────┬─────┬─────┬─────┬───┬─┐
│1│2 2│3 3 3│4 4 4│5 5 5│6 6 6│7 7 7│8 8│9│
└─┴───┴─────┴─────┴─────┴─────┴─────┴───┴─┘

   ] +//. 3 ]\ a 
1 4 9 12 15 18 21 16 9


Try it online!