EQUIPMENT

Starting in an athletic stance and finishing athletic "shoes help"

Transitioning from crosstraining shoes to weightlifting shoes can help with keeping the torso up right in both lift off position and the catch positon in the snatch and clean. If you have suspect ankle mobility then shoes will help create a strong and stable position for you while practicing the lifts. If do a full squat meaning you lower your hips below the knees without the heels coming up or torso leaning over, then you have healthy ankle mobility. Ankle, hip and shoulder mobility all work in unison for the squat and the snatch & clean and jerk. So if your heels come up in a low squat then Weightlifting shoes will help you place your entire foot on the ground which will help you become stronger from start to finish in a squat or a snatch and clean. In order to catch a snatch or clean consistently our torso’s must be somewhat upright if not the torso lean will push the bar away from us. Weightlifting shoes also have a hard sole which leads to a stronger lift off and a stronger squat. Force from body goes into moving the body then that will translate right into the bar. Running shoes are cushioned to soften the impact of the feet hitting the ground. Squating with running shoes decreases force production by the softness or give that running shoes are created to do when there is force placed on the feet through the body.
One more side note. Landing balanced and athletic is important for the snatch and clean. When learning the lifts or if you are a lifter that over pulls can lead to our bodies falling back on out heels. New lifters want to through or leverage the body into the bar which then creates a slight loop in the pull when this happens there is a small decrease in the pull, when this happen our bodies weight shifts backwards. This happens a lot with flat soled shoes and new lifters. You can lift in cross trainers but you must stay athletic from start to finish. This will straighten out your bar path too.
These are just a couple of quick tid bits on what weightlifting shoes help with and also how to lift better in flat soles shoes if you are a cross fitter.