We shall build the cosmological perturbations to a unified dark fluid with sharp transition in the late-time universe. The model evolves from a CDM-like behavior to a quasi-$\Lambda$ phase in which the equation of state parameter is $w = -1 + \epsilon$, with $\epsilon < 1$ a constant. We show that the power spectrum associated to this model exhibits a growing mode scaling with hyperbolic functions of the scale k and the conformal time. This mode is dangerous for causing unphysical instabilities in the over-densities. The growing mode can be eliminated if the redshift of transition between the CDM regime and the quasi-$\Lambda$ phase is nearly zero.