The easiest way to do this, IMO:
1) give your text object an “align to spline” tag and use the path spline from your sweep;
2) animate the sweep end growth as normal;
3) right click on the words “end growth” in the attributes manager and from the context menu, select Animation>Set Driver;
4) now select the align to spline tag you created above, right click on the word, “position”, and from the context menu select Animation>Set Driven (absolute).
This sets up an xpresso for you that will drive the align to spline tag’s position with the sweep’s growth.