The present work addresses the question of a nonhorizontal coexistence plateau found in the liquid-expanded (LE) to liquid-condensed (LC) transition of Langmuir monolayers of lipid amphiphiles, which is apparently incongruent with the first-order character of this main LE/LC phase transition. This pathology is understood in a mechanical context as a resistance of the monolayer against compression giving rise to a nonzero rigidity in the coexistence region. Surface rheology has allowed for a quantitative determination of the compression parameters, namely, dilational elasticity epsilon and viscosity eta. Data for the phase coexistence region reveal dynamical stiffening at faster deformation, which points out a chief control of lipid diffusion on monolayer rigidity. Monolayer viscosity remains however low at the value corresponding to the continuous fluid phase. The presence of coexistence domains is then invoked as the structural element responsible for such a nontrivial rheology, the finite domain growth rate imposing a kinetic limit for equilibrium compression along a quasi-static path. Brewster angle microscopy has allowed for studying the kinetic mechanism for domain growth. The finite rigidity observed at the coexistence region is related to the resistance of LC domains to grow at the expense of the LE phase. A reconciliation of the nonhorizontal plateau observed at finite compression rates with the first-order character of the LE/LC transition emerges then naturally from this kinetic scenario. New mechanical features are consequently assigned to the phase-separated monolayers made of stiff grains rafting in a fluid matrix. Particularly, for a raft-like lipid distribution, we hypothesize a finite rigidity kinetically controlled by the rate of domain growth and a high fluidity controlled by the continuous phase. We have depicted a minimal model of membrane mechanics that accounts for the elasticity of such a heterogeneous composite medium. This "Plum-cake" model is able to qualitatively predict the observed mechanical features and is suggested to describe raft-like membranes as compliant elastic media where lipid domains work as reservoirs able to exchange material with the surrounding fluid phase.