Due to the rise of user-generated content, social media is increasingly adopted as a channel to deliver customer service. Given the public character of these online platforms, the automatic detection of emotions forms an important application in monitoring customer satisfaction and preventing negative word-of-mouth. This paper introduces EmoTwiCS, a corpus of 9,489 Dutch customer service dialogues on Twitter that are annotated for emotion trajectories. In our business-oriented corpus, we view emotions as dynamic attributes of the customer that can change at each utterance of the conversation. The term 'emotion trajectory' refers therefore not only to the fine-grained emotions experienced by customers (annotated with 28 labels and valence-arousal-dominance scores), but also to the event happening prior to the conversation and the responses made by the human operator (both annotated with 8 categories). Inter-annotator agreement (IAA) scores on the resulting dataset are substantial and comparable with related research, underscoring its high quality. Given the interplay between the different layers of annotated information, we perform several in-depth analyses to investigate (i) static emotions in isolated tweets, (ii) dynamic emotions and their shifts in trajectory, and (iii) the role of causes and response strategies in emotion trajectories. We conclude by listing the advantages and limitations of our dataset, after which we give some suggestions on the different types of predictive modelling tasks and open research questions to which EmoTwiCS can be applied. The dataset is available upon request and will be made publicly available upon acceptance of the paper.