Despite being prominent and ubiquitous, message-based interaction is limited in nonverbally conveying emotions. Besides emoticons or stickers, messaging users continue seeking richer options for affective communication. Recent research explored using chat balloons' shape and color to communicate emotional states. However, little work explored whether and how chat-balloon animations could be designed to convey emotions. We present the design of AniBalloons, 30 chat-balloon animations conveying Joy, Anger, Sadness, Surprise, Fear, and Calmness. Using AniBalloons as a research means, we conducted three studies to assess the animations' affect recognizability and emotional properties (N = 40), and probe how animated chat balloons would influence communication experience in typical scenarios including instant messaging (N = 72) and chatbot service (N = 70). Our exploration contributes a set of chat-balloon animations to complement non-nonverbal affective communication for a range of message-based interfaces, and empirical insights into how animated chat balloons might mediate particular conversation experiences (e.g., perceived interpersonal closeness, or chatbot personality).