We report an experimental and theoretical investigation of the entry and passage behaviour of biological cells (HeLa and MDA-MB-231) in a constricted compliant microchannel. Entry of a cell into a micro-constriction takes place in three successive regimes: protrusion and contact (cell protrudes its leading edge and makes a contact with the channel wall), squeeze (cell deforms to enter into the constriction) and release (cell starts moving forward). While the protrusion and contact regime is insensitive to the flexibility of the channel, the squeeze zone is significantly smaller in the case of a more compliant channel. Similarly, in the release zone, the acceleration of the cells into the microconstriction is higher in the case of a more compliant channel. The results showed that for a fixed size ratio ρ and E_c, the extension ratio λ decreases and transit velocity U_c increases with increase in the compliance parameter f_p. The variation in the cell velocity is governed by force due to the cell stiffness F_s as well as that due to the viscous dampening F_d, explained using the Kelvin–Voigt viscoelastic model. The entry time t_e = m(ρ)^k₁(1 + f_p)^k₂(E_c)^k₃ and induced hydrodynamic resistance of a cell ΔR_c/R = k(ρ)^a(1 + k_ff_p)^b(k_EE_c)^c were correlated with cell size ratio ρ, Young's modulus E_c and compliance parameter f_p, which showed that both entry time t_e and the induced hydrodynamic resistance ΔR_c are most sensitive to the change in the compliance parameter f_p. This study provides understanding of the passage of cells in compliant micro-confinements that can have significant impact on mechanophenotyping of single cells.