Not exactly "best walking shoes" but I thought I'd share my experience replacing my shoe laces with stretch versions. I had one particular pair of lace up shoes I liked but they constantly came untied no matter what I did. I saw one brand of stretch laces - Hickies I think - but thought they were way too expensive. Also I didn't like that they just made the shoes slip on but didn't give any room to pull the shoe open wider while putting them on or accommodated different sock thicknesses.
Ultimately bought a different type from Amazon- stretchy cord with a spring locking gizmo and snap on end caps. You thread the laces just like normal, trim off the excess cord then slide on the spring lock and the end caps. I leave about 2 inches excess but you can experiment and trim off more if needed. They come in different brands and colors on Amazon for about $6-7 a pair.