If every action in your text object is followed by a line break (newline expression), then you can remove the oldest lines from the beginning of the text before adding new lines.
For example, your text object can display 4 lines.
When you need to add the 5th line, remove 1st line from the text. You can do this using find(text.text, newline) or tokenat/tokencount expressions.
aaa bbb bbb
bbb -> ccc -> ccc
ccc ddd ddd
Alternatively, you can try rex's "Text Scrolling" plugin.