In LiveType, you would set up your text and logo in the postions, size, color etc that you want.
Then use the track grouping controls and apply a Scroll effect. Because the text and logo elements are grouped, both move together.
In Final Cut Pro, you would need to nest the text and logo clips, then apply a global move to the nest.
Another way is to use Photoshop. Start with one of the video templates that fits the pixel dimensions of your FCP Sequence. Extend the PSD canvas size vertically -doubling or tripling its height to make space for all your elements. Continue to extend downwards as needed.
When everything is in position, turn off the background, merge visible to send your text and logo to one layer, export as TIFF with transparency enabled.
Bring that into FCP and use the Motion parameters to move it up the screen.