Aims. The role of minor galaxy mergers in galaxy evolution, and in particular to mass assembly, remains an open question. In this work we measure the merger fraction, f m , of L B > ∼ L * B galaxies in the VVDS-Deep spectroscopic survey, and study its dependence on the B-band luminosity ratio the pair galaxies, μ ≡ L B,2 /L B,1 , focusing on minor mergers with 1/10 ≤ μ < 1/4, and on the rest-frame NUV − r colour of the principal galaxies. Methods. We use spectroscopic pairs with redshift z < ∼ 1 in the VVDS-Deep survey to define kinematical close pairs as those galaxies with a separation on the sky plane 5 h −1 kpc < r p ≤ r max p and a relative velocity Δv ≤ 500 km s −1 in redshift space. We vary r max p from 30 h −1 kpc to 100 h −1 kpc. We study f m in two redshift intervals and for several values of μ, from 1/2 to 1/10. We take μ ≥ 1/4 and 1/10 ≤ μ < 1/4 as major and minor mergers. Results. The merger fraction increases with z and its dependence on μ is well described by a power-law function, f m (≥μ) ∝ μ s . The value of s evolves from s = −0.60 ± 0.08 at z = 0.8 to s = −1.02 ± 0.13 at z = 0.5. The fraction of minor mergers for bright galaxies shows little evolution with redshift as a power-law (1+z) m with index m = −0.4±0.7 for the merger fraction and m = −0.5±0.7 for the merger rate, in contrast with the increase in the major merger fraction (m = 1.3 ± 0.5) and rate (m = 1.3 ± 0.6) for the same galaxies. We split our principal galaxies in red and blue, finding that i) f m is higher for red galaxies at every μ, ii) f red m does not evolve with z, with s = −0.79 ± 0.12 at 0.2 < z < 0.95, and iii) f blue m evolves dramatically: the major merger fraction of blue galaxies decreases by a factor of three with cosmic time, while the minor merger fraction of blue galaxies is roughly constant. Conclusions. Our results show that the mass of normal L B > ∼ L * B galaxies has grown by about 25% since z ∼ 1 because of the combined effects of minor and major mergers. The relative contribution of the mass growth by merging is ∼25% due to minor mergers and ∼75% due to major mergers. The relative effect of merging is more important for red than for blue galaxies, with red galaxies subject to 0.5 minor and 0.7 major mergers since z ∼ 1, which leads to a mass growth of ∼40% and a size increase by a factor of 2. Our results also suggest that, for blue galaxies, minor mergers likely lead to early-type spirals rather than elliptical galaxies. These results show that minor merging is a significant but not dominant mechanism contributing to the mass growth of galaxies in the last ∼8 Gyr.