Hi M.,
It may simply be gas. If you've had a C-section you would know that gas can even be in a shoulder blade! Kidney stone pain can move but it is usually in the back area and most pain comes from a stone lodging so you normally don't feel it moving. If you do, it's almost a relief....I've had numerous ones over my lifetime so memories are vivid.
The pain you are describing sounds like ovulation. These feelings come and go with the blood flow and go from side to side because you can feel it in both the ovaries. Is that the area you mean when you say lower left abdomen? Check your calendar and see if it coincides. If it doesn't you may need to consult a diagnostician.
God bless and I hope you feel better!
M.