We study quasi-periodic transformations between nonlocal spatial solitons of different symmetries triggered by modulational instability and resembling a self-induced mode converter. Transformation dynamics of solitons with zero angular momentum, e.g. the quadrupole-type soliton, reveal the equidistant spectrum of spatial field oscillations typical for the breather-type solutions. In contrast, the transformations of nonlocal solitons carrying orbital angular momentum, such as 2x3 soliton matrix, are accompanied by their spiralling and corresponding spectra of field oscillations show mixing of three characteristic spatial frequencies.