We study imbibition of a monodisperse emulsion into high-aspect ratio microfluidic channels with the height h comparable to the droplet diameter d. Two distinct regimes are identified in the imbibition dynamics. In a strongly confined system (the confinement ratio d/h = 1.2 in our experiments), the droplets are flattened between the channel walls and move more slowly compared to the average suspension velocity. As a result, a droplet-free region forms behind the meniscus (separated from the suspension region by a sharp concentration front) and the suspension exhibits strong droplet-density and velocity fluctuations. In a weaker confinement, d/h = 0.65, approximately spherical droplets move faster than the average suspension flow, causing development of a dynamically unstable high-concentration region near the meniscus. This instability results in the formation of dense droplet clusters, which migrate downstream relative to the average suspension flow, thus affecting the entire suspension dynamics. We explain the observed phenomena using linear transport equations for the particle-phase and suspension fluxes driven by the local pressure gradient. We also use a dipolar particle interaction model to numerically simulate the imbibition dynamics. The observed large velocity fluctuations in strongly confined systems are elucidated in terms of migration of self-assembled particle chains with highly anisotropic mobility.