Polydeuces

Strike the joints simply means take a smooth object and tool the joint smooth. It compresses the mortar and brings up any water. The compression is most important, watch a pointing video to get an idea, very simple really very quick. Just a few strokes to top dress.

I have had good results with AP Green Green patch 421 which is used as a brick joint mortar as well as for patching. That appears to be an oval kiln and the joint cracks I would guess from flexure. In my experience, I think I would patch all smoothly and before the patch sets and cures, tighten the lid band until nice and snug to its final tension. Final strike all the joints even though they are small. I would also wipe away excess patch to keep this as smooth and thin as practical.
Cleaning and vacuuming the grooves first is important as well as reasonably pre-wetting the joints so the material has time to set especially if warm.
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Thanks, Neil! They don't go all the way through as of yet. I'm wondering if it might be the mud daubers around here or something, but I've never noticed any critters hanging out on top of it.

it looks like someone is scratching it out of the seams.  the distance from the top of the kiln down to it's  bottom is in inches.    the crack looks like it could be measured in significantly less than one inch.    probably not a worry unless you see it coming through the bottom.   kilns do not have to be kept pristine, just reasonably clean.