Britain has dramatically increased the value of weaponry and defence equipment it sells to the world’s most repressive regimes since vows by senior ministers to expand arms exports after the Brexit vote.

For a quarter of a century, water, land and ethnic conflicts have poisoned ties between the five ’stans. Now, even as the U.K. pulls away from the European Union, and other countries in the bloc mull their future in it, Central Asian nations are opening up to one another, taking steps to establish what in a few years could amount to their version of a Schengen zone.