I can't really see it that well, but you can use mortar mix to fill holes in stone walls and cracks in the walkway.
Just clean them out and make a little wider if needed.
They actually make product that comes in caulking tube but is a flexible mortar mix. Usually in the aisle near trowels and other masonry products at HD or Lowes. I think Sacrete makes it.
Fix this before your basement leaks. Suggest you get product advice from Orange Hardware. They were very helpful to me in a different pavement problem that Home Depot could not help well.
Thanks all - we have had issues on that side of the house - there is a sump pump on that side with the house already, just try to minimize it's use.
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What's the best way to fill these in? Looks the previous fix was a black rubber caulk - or a peppble concrete mixture.