Seeds are getting in there, germinating, and growing. I have tried many types of mulch material over the years and found that even stones get dust and dirt particles accumulating in them and provide a place for the seeds that also get in those spaces to germinate. Diligent inspection and taking care of the unwanted plant growth is the only sure fire way to keep that growth down.

kimmq is kimmsr

It looks like such a difficult place to get into to weed. Could you consider a low-maintenance groundcover that would smother any weed for that space? Anything from pachysandra to lily of the valley to nepeta? Or ivy, and let it cover the chain link as well?

I tried for years to kill a Mimosa growing in a fence like that. I removed the foliage every day (that it wasn't raining, or I wasn't home) but it never gave up by the time I moved away from it, 4 yrs after I decided to "kill" it that way. There are some circumstances that just require more serious measures. You could cut that as low to the ground as possible, use a toxin on the cut surface before it heals over. That strip between the fences looks like it should just be concrete. If a trimmer won't fit between them, the alternatives are to reach your hand through to pull whenever you see something sprouting, if your hand fits, or use some kind of chemical as an ongoing maintenance plan.