Millimeter wave (mmWave) cell-free MIMO achieves an extremely high rate while its beam alignment (BA) suffers from excessive overhead due to a large number of transceivers. Recently, user location and probing measurements are utilized for BA based on machine learning (ML) models, e.g., deep neural network (DNN). However, most of these ML models are centralized with high communication and computational overhead and give no specific consideration to practical issues, e.g., limited training data and real-time model updates. In this paper, we study the {probing} beam-based BA for mmWave cell-free MIMO downlink with the help of broad learning (BL). For channels without and with uplink-downlink reciprocity, we propose the user-side and base station (BS)-side BL-aided incremental collaborative BA approaches. Via transforming the centralized BL into a distributed learning with data and feature splitting respectively, the user-side and BS-side schemes realize implicit sharing of multiple user data and multiple BS features. Simulations confirm that the user-side scheme is applicable to fast time-varying and/or non-stationary channels, while the BS-side scheme is suitable for systems with low-bandwidth fronthaul links and a central unit with limited computing power. The advantages of proposed schemes are also demonstrated compared to traditional and DNN-aided BA schemes.