I would like to use a parameter behaviour to move a layer instead of keyframes. Problem is that I would like a curve that flattens a lot more than "ramp" does. I have also tried with "logarithmic" - that doesn't really look right either.
Is there away to make a parameter behaviour control the x-axis with more fine control over the curve?
You can use a null object and link it the X position parameter of your object.
There are a few different things in Motion that give you access to not only easing, but also Accelerate and Decelerate. These options are available in Sequence Replicators as well as Sequence Text, both of which would probably be hard to adapt.
One behavior that has these options is Motion Path... If you set up a null object on a Motion Path, you can control the x-axis motion of your object by paralleling it with the null object and Motion Path. Create a Open Spline path that curves upward at the end. Link the position X parameters of your null and object to be animated. You can control/customize the amount of time it takes for the null to travel vertically as the slowing to a stop of the linked object. The longer the "journey" the better the decrease in speed of your main object.
Set the Speed on the Motion Path to Custom. Use the Ramp behavior on the Motion Path Custom Speed. The benefit is that you can set the start and end times with the ramp behavior. (you might want to reset the Custom Speed to "clear" the default keyframed settings first but it might not be necessary. [It just annoys me that Custom Speed has default keyframes and it's almost never what I want.])