Applications of intelligent reflecting surfaces (IRSs) in wireless networks have attracted significant attention recently. Most of the relevant literature is focused on the single cell setting where a single IRS is deployed and perfect channel state information (CSI) is assumed. In this work, we develop a novel methodology for multi-IRS-assisted multi-cell networks in the uplink. We consider the scenario in which (i) channels are dynamic and (ii) only partial CSI is available at each base station (BS); specifically, scalar effective channel powers from only a subset of user equipments (UE). We formulate the sum-rate maximization problem aiming to jointly optimize the IRS reflect beamformers, BS combiners, and UE transmit powers. In casting this as a sequential decision making problem, we propose a multi-agent deep reinforcement learning algorithm to solve it, where each BS acts as an independent agent in charge of tuning the local UE transmit powers, the local IRS reflect beamformer, and its combiners. We introduce an efficient information-sharing scheme that requires limited information exchange among neighboring BSs to cope with the non-stationarity caused by the coupling of actions taken by multiple BSs. Our numerical results show that our method obtains substantial improvement in average data rate compared to baseline approaches, e.g., fixed UE transmit power and maximum ratio combining.