Crossing national borders — existing in, or working across, more than one country. Used most often for companies, organisations, movements, crime networks and problems that do not stop at a border.

Here is the useful distinction, and it is worth learning properly. 'International' means BETWEEN nations, and quietly keeps the nations as the main actors: an international agreement is nations dealing with each other. 'Transnational' means ACROSS nations, and suggests the borders barely matter to the thing being described: a transnational company does not really feel like a company from one country doing business abroad — it is spread across many, with staff, money and offices everywhere, loyal to no single flag. Writers choose 'transnational' precisely when they want that flavour: this thing has escaped the country box.

So you see it a lot with: transnational corporations, transnational crime, transnational networks, transnational families (family members living in different countries but still one family), transnational identity.

Register: academic, journalistic, policy-writing. It is a serious word from serious pages — you will read it in the news or a textbook far more often than you will hear it in a café. It is an adjective; the noun 'transnational' meaning 'a transnational company' also exists but is much rarer.