Abstract: While offering significant promise for diverse applications, pattern-oriented swarms encounter multifaceted challenges in geometric control, self-organization, and safe navigation through dynamic environments. In this paper, we present a GRF-based stochastic optimal control framework to address these challenges within a unified probabilistic architecture. By extending the GRF into the temporal domain, the proposed framework casts collective coordination as a Bayesian inference task, enabling swarms to accommodate environmental uncertainty, satisfy non-convex constraints, and reconcile heterogeneous dynamics across diverse platforms. We develop an uncertainty- and safety-aware collision avoidance module for navigation in the presence of stochastic obstacle motion. The unscented transform is employed to propagate state uncertainty for both dynamic obstacles and neighboring agents, yielding principled confidence bounds for collision avoidance. In addition, density-guided pattern control is introduced, which encodes geometric patterns as implicit density fields. This representation decouples pattern specification from explicit agent-to-target assignments, thereby facilitating intrinsic self-healing and elastic reconfiguration in a distributed manner. The proposed framework is extensively evaluated through Monte Carlo simulations across diverse scenarios. Its model-agnostic nature is demonstrated on both quadrotor and fixed-wing UAV swarms, highlighting its generalizability across platforms with heterogeneous dynamics. Finally, the efficacy and robustness of the proposed method are validated through indoor experiments with a 15-quadrotor swarm and outdoor deployments involving 4 custom-built autonomous quadrotors. These experiments substantiate the proposed framework's capacity to maintain reliable geometric pattern transitions and safety-aware navigation within real-world environments.