Answer first
For travel, decant shampoo and tonic into small leak-proof bottles within airline liquid limits, pack topicals upright in a zip bag, and keep to your usual times. Hotel shampoos and hard water can feel harsh, so a familiar mild shampoo helps. Wear a hat in strong sun.
Packing
Small bottles, zip bags, caps taped if needed.
On arrival
Rinse off chlorine or sea water soon after swimming.
Questions
Can I bring the full bottles?
In checked luggage, yes.
What about humid destinations?
Wash more often if your scalp gets oily.