Despite the rapid advance in multispectral (MS) pansharpening, existing convolutional neural network (CNN)-based methods require training on separate CNNs for different satellite datasets. However, such a single-task learning (STL) paradigm often leads to overlooking any underlying correlations between datasets. Aiming at this challenging problem, a multitask network (MTNet) is presented to accomplish joint MS pansharpening in a unified framework for images acquired by different satellites. Particularly, the pansharpening process of each satellite is treated as a specific task, while MTNet simultaneously learns from all data obtained from these satellites following the multitask learning (MTL) paradigm. MTNet shares the generic knowledge between datasets via task-agnostic subnetwork (TASNet), utilizing task-specific subnetworks (TSSNets) to facilitate the adaptation of such knowledge to a certain satellite. To tackle the limitation of the local connectivity property of the CNN, TASNet incorporates Transformer modules to derive global information. In addition, band-aware dynamic convolutions (BDConvs) are proposed that can accommodate various ground scenes and bands by adjusting their respective receptive field (RF) size. Systematic experimental results over different datasets demonstrate that the proposed approach outperforms the existing state-of-the-art (SOTA) techniques.