In everyday makeup, doing foundation & concealer first works best. Professional makeup artists usually prefer to do eyes first.
Foundation and concealer should not be used on eyelids, only an eye primer. Primer is also applied directly under bottom lashes to keep liner from moving. Concealer on lids is oily and can crease. Another factor to consider is having foundation on the face means wiping it off the brows to get a matte, makeup free surface to apply color and see the true shade and if it's natural looking.
During application, many shadow shades are blended, mixed and taken off until the color seems right. Using eye makeup remover would disturb any other face makeup. Sometimes for photos, I'll do lips and brows first to give me balance on the face and a map to begin work. That done, often less shadow, blush or bronzer is needed.
Doing my own makeup, skincare/foundation and concealer comes first and then the eye makeup. If using more shadow than usual, I just place a Kleenex lightly under the eye to keep fallout from landing on corrector/concealer and foundation. It also gives those products time to settle into the skin.
After shadow is brows, liner and mascara. Then bronzer and blush, lips and gloss. Last step is setting spray.
I rarely do a full makeup on myself. I use tinted moisturizer, concealer for areas that need extra coverage, cream blush, brows, mascara and tinted lip balm. Less is more for me. I like a dewy, more youthful look without a lot of time involved. I can build on that basic look at any time.