The magnetized liner inertial fusion (MagLIF) approach to inertial confinement fusion [Slutz et al., Phys. Plasmas 17, 056303 (2010); Cuneo et al., IEEE Trans. Plasma Sci. 40, 3222 (2012)] involves subsonic/isobaric compression and heating of a deuterium-tritium plasma with frozen-in magnetic flux by a heavy cylindrical liner. The losses of heat and magnetic flux from the plasma to the liner are thereby determined by plasma advection and gradient-driven transport processes, such as thermal conductivity, magnetic field diffusion, and thermomagnetic effects. Theoretical analysis based on obtaining exact self-similar solutions of the classical collisional Braginskii's plasma transport equations in one dimension demonstrates that the heat loss from the hot compressed magnetized plasma to the cold liner is dominated by transverse heat conduction and advection, and the corresponding loss of magnetic flux is dominated by advection and the Nernst effect. For a large electron Hall parameter ( ω e τ e ≫ 1 ), the effective diffusion coefficients determining the losses of heat and magnetic flux to the liner wall are both shown to decrease with ω e τ e as does the Bohm diffusion coefficient c T / ( 16 e B ) , which is commonly associated with low collisionality and two-dimensional transport. We demonstrate how this family of exact solutions can be used for verification of codes that model the MagLIF plasma dynamics.