Cooperative mission planning for heterogeneous teams of mobile robots presents a unique set of challenges, particularly when operating under communication constraints and limited computational resources. To address these challenges, we propose the Cooperative and Asynchronous Transformer-based Mission Planning (CATMiP) framework, which leverages multi-agent reinforcement learning (MARL) to coordinate distributed decision making among agents with diverse sensing, motion, and actuation capabilities, operating under sporadic ad hoc communication. A Class-based Macro-Action Decentralized Partially Observable Markov Decision Process (CMacDec-POMDP) is also formulated to effectively model asynchronous decision-making for heterogeneous teams of agents. The framework utilizes an asynchronous centralized training and distributed execution scheme, enabled by the proposed Asynchronous Multi-Agent Transformer (AMAT) architecture. This design allows a single trained model to generalize to larger environments and accommodate varying team sizes and compositions. We evaluate CATMiP in a 2D grid-world simulation environment and compare its performance against planning-based exploration methods. Results demonstrate CATMiP's superior efficiency, scalability, and robustness to communication dropouts and input noise, highlighting its potential for real-world heterogeneous mobile robot systems. The code is available at https://github.com/mylad13/CATMiP