Photoelectrical response in CuInS2-ZnIn2S4 alloys single crystals was analysed in the lowtemperature region from 30 K up to 100 K. The molar ratio of ZnIn2S4 in the alloys was varied in the range 0 mol%-16 mol%. The crystals with up to 12 mol% were the single-crystalline, meanwhile those with 16 mol% were the two phase ones. We have analysed spectral distribution of their photocurrent at different temperatures and the following relaxation towards the stationary values. The photo-induced photoconductivity phenomena were identified. Moreover the long-lasting relaxations with characteristic times exceeding 1.5x10(3) sec were observed at lowest temperatures. They used to shorten exponentially with increasing temperature showing thermally activated behaviour. The main parameters of the photoconductivity kinetics and their temperature dependencies were determined. The observed behaviour was explained by the slow multicenter recombination due to the combined effect of different trapping and recombination centers. The effects of both - "fast" and "slow" recombination centers were taken into account.