Depending on how many 'lines' you need I believe it's entirly do-able in Premiere or AE.

In premiere you could use the titler and just adjust motion/postion as needed to move each line. Overlapping each as neccesary. There are many in-depth masking features that you can use to have text being and end on a 'bar that moves across a screen or text that lops in this fashion.

For AE it's much the same but import each still line as a layer and parent them with your text, it can be a little more daunting but once you get used to the functions you'll be using in AE it becomes a snap.

Since I'm not entirly sure what you're looking for I can only suggest basic things. If you see some footage that mimics what you want or close, pop us a link so we can give you a bit better informed idea.