Example of a non-Kähler manifold with varying plurigenera

The first example of this phenomenon was discovered by Iku Nakamura in his 1975 paper Complex parallelisable manifolds and their small deformations, J. Differential Geom. 10 (1975), 85-112.

The manifold $X$ is a $3$-dimensional solvmanifold, and he writes down its small deformations explicitly, see his Theorem 2 and the discussion on pages 96-99.