Accurate long-term localization using onboard sensors is crucial for robots operating in Global Navigation Satellite System (GNSS)-denied environments. While complementary sensors mitigate individual degradations, carrying all the available sensor types on a single robot significantly increases the size, weight, and power demands. Distributing sensors across multiple robots enhances the deployability but introduces challenges in fusing asynchronous, multi-modal data from independently moving platforms. We propose a novel adaptive multi-modal multi-robot cooperative localization approach using a factor-graph formulation to fuse asynchronous Visual-Inertial Odometry (VIO), LiDAR-Inertial Odometry (LIO), and 3D inter-robot detections from distinct robots in a loosely-coupled fashion. The approach adapts to changing conditions, leveraging reliable data to assist robots affected by sensory degradations. A novel interpolation-based factor enables fusion of the unsynchronized measurements. LIO degradations are evaluated based on the approximate scan-matching Hessian. A novel approach of weighting odometry data proportionally to the Wasserstein distance between the consecutive VIO outputs is proposed. A theoretical analysis is provided, investigating the cooperative localization problem under various conditions, mainly in the presence of sensory degradations. The proposed method has been extensively evaluated on real-world data gathered with heterogeneous teams of an Unmanned Ground Vehicle (UGV) and Unmanned Aerial Vehicles (UAVs), showing that the approach provides significant improvements in localization accuracy in the presence of various sensory degradations.