The most common solvent for NMR experiments is deuterated chloroform, which is a carcinogen. So you do not want to transport your NMR tubes in your shirt pocket, your handbag or other places, where it can easily drop and break.
Pls use a secondary containment for transporting your prepared samples from your lab to the NMR lab. At minimum, you should place your sample in an Erlenmeyer Flask, which contains some absorbent medium. A good absorbent would be vermiculite. Chances are that you have plenty of that in your lab since this is used as protective layer for the shipping of chemicals. The best solution for the transport of NMR samples are “NMR tube carriers”, which are shatter and solvent resistant polymer tubes. You can purchase those from the major vendors of NMR tubes and a search in workday will point you to several vendors.