You might consider filling the entire in question w/sand. Yes, sand. It should be able to handle any water that manages to filter down.
Broom it down and even water it so it settles COMPLETELY. If you don't do this (in spite of running the risk of water getting in the house), the sand will settle on it's OWN (over time), thus, creating a cavity (below your sealer.)
The Sealer I'm refering to is Concrete Caulk. Once you've filled and managed to 'settle' the sand to within 1/2" of the TOP LEVEL of the walkway, start using the Concrete Caulk. Be SURE the caulk is 'rated' for use in openings .5", (give or take since your crack may be wider in places, or narrower.) If you can't find what you need at the Home Center, visit perhaps a Stone Quarry, or a Cement Facility, or a Concrete Facility, or anything along those lines. There are even cataloge products that we are 'offered' here at work from time to time. So, they're out there. Use the Yellow Pages for starters, and start looking for these business under the 'headings' I've given or something similiar.
My best to ya and hope this helps.
Jay J -Moderator
PS Since your walkway can 'move' over the years, you may have to keep up w/this from time to time. You'd do as you're doing now - Clean out the 'affected areas', and re-do. Also, consider having gutters installed (if water runs off the roof, and you'll only know this FOR SURE if you're out there, looking up, during a rainstorm.) Consider some 'edging' on the 'other side' of the walkway, i.e., the side that's NOT up against the house, to keep any runoff from running across the walkway and towards the wall. I guess you'll know how good of a job it is you did when you find 'puddles' where the walkway meets the house. OK - Now what do you do??! Good question! In WInter, it will reek havoc on that area. If it's just a little water, keep your eyes on it to see if it's evaporating or 'leaching' down again. At worst, you MAY have to rip up the walkway and do it again, this time making sure it slopes AWAY from the foundation. BE SURE you have a good 'base' too if it comes to this. (And maybe then you can fill the hole in the foundation.) OH, tar and cement don't 'bond' well, so I'd stay away from that 'material' UNLESS the Container specifically says it will 'bond' with concrete. And lastly, you say 'old stone blocks'. I'm assuming cement blocks. If you're refering to SOLID STONE, you'll probably want to talk to a Foundation Contractor about this first. They are the ones that do this. If you can't find such a contractor, ctc. your local Building Inspector for leads. Or, visit almost any Architect or Civil Engineer for leads. Believe me - These folks deal with ALL kinds of Contractors.