Elements off screen not moving

Here's how I would approach the problem. 

1. Every element and all of the layers and any animations that will move from the bottom of the comp frame to the top would be a separate composition. Name these comps something like Slide 1, Slide 2, and Slide 3. If they are going to be visible for 2 seconds and then move up for 1 second, make the comps 3 seconds long.  Another option would be to make the comps for each slide about 7 seconds long and then automate the animation with an animation preset.
2. When you have all of the slide comps created, select them all in the Project Panel and then drag them to the New Comp icon or right-click or use the File menu to Create a new Comp from selected and choose One comp, sequence layers, and overlap for the duration of the move.
3. Animate the first nested comp's position so that it moves onscreen, then off, and save the move as an animation preset.  
4. Select each additional layer and apply the animation preset. 

If you use an expression based on the layer in and out point, you can just select all layers and apply the preset once. In about two minutes, properly prepared nested comps can be turned into your final animation.

If you need the project to start with the first slide in place, just trim the work area's in-point and set the Comp to the Work Area or just render the work area. 

Here's the expression I would use to automate a slide in from the bottom, pause, then slide out. 

movTime = thisComp.frameDuration * 24; // move time in frames
ip = thisComp.height + height/2;
rp = thisComp.height/2;
ep = 0 - height/2;
t = time;
M1 = ease(t, inPoint, inPoint + movTime, ip, rp);
M2 = ease(t, outPoint -movTime, outPoint, rp, ep);
if (t - inPoint < movTime)
	p = [thisComp.width/2, M1];
else
	p = [thisComp.width/2, M2];

You can change the ease to easeOut or easeIn. Control the spacing by the overlap. 

I've included the Project file.