When I travel with soapy stuff, this is what I do to prevent spillage, etc, as the car bounces around the highways as well as changing altititudes:
Tightly close lids on all bottles. Tape the outside of the lid-to-bottle juncture to disouage leakage, but this doesn't work with vinegar. Seal opened, but tightly sealed liquids in zip-lock baggie prior to packing UPRIGHT in a box that will not allow them to fall over. Keep the bottles of liquids upright during transport.
Seal lye containers with lids tightly closed, inside double plastic bags, double sealed. Other powdery substances are also well sealed in zip-lock baggie as needed and packed inside other containers I need for soaking, such as molds, a crockpot, bowls, etc. Packing soft chemicals inside breakable is one way to cushion them, as well as space efficient.
As for soap I make while traveling, which I cut and then had to transport back home, this is what I have done in the past:
Individually wrap each bar in packing paper, which is a bit more substantial than newsprint, but not as stiff as a grocery bag. Other times, when I didn't have any paper like that, I have used tissue paper, and I have also used cut up cotton jersey fabric (like t-shirt material). The purpose is to prevent soaps from humidity as well as to prevent damage during packing and transport. Then pack them in shoe boxes, boot boxes, or other available cardboard boxes I may have or easily find at a grocery store (before the grocer cuts them down to send to recycle.) Sometimes I use other containers, such as plastic baskets, etc. when I happen to have one with me.