Saudi Arabia’s crown prince is embarking on what is expected to be a three-country sweep through Asia to nurture alliances there as the murder of a Saudi dissident and the Saudi-led war in Yemen continue to disrupt the kingdom’s relations with the U.S. and European powers.

Angered by Western criticism, Prince Mohammed bin Salman has told aides that he would seek diplomatic support from Asian powers including China and India, according to senior Saudi government advisers.

The...