Can cosmetics with pump head be shipped by air?

You mean by express? Or take it on the plane yourself? You'd better find a company to ask if it can be delivered by courier. If you bring your own on the plane, you can bring no more than 100ml of liquid with you, and liquids with a volume between 100ml and 1L can be put in the suitcase for use. Each airline is somewhat different, it is best to check the official website.