Each text bubble is duplicated from a rigged template speech bubble. The rig controls the dimensions of the bubble based on the size of the text it contains. Drivers are heavily used on the bones to coordinate them with other bones and movement.

Each bubble is parented to a screen object which scrolls upwards, but the parenting is dynamic, so that it only engages when the message has been "sent".

The "typewriter effect" of each bubble is not keyframed (unsupported) but handled by a frame update callback. The typing is determined in advance with random jitters to emulate a person typing.

Hope that explains it! I have considered sharing the framework, but I know I will get a lot of requests that I'm not prepared to handle unless I was receiving donations.