I would like to link an object's position to a slider controller but i want to follow this slider with ease.
Here is a gif for more detail where's the red rectangle is the object the white dot is the slider:
Any tips how can I do it or at all it is possible?
Perhaps I don't understand what you want to do, but the slider is an interactive thing done before you hit render to set a value, and the animation only happens once you hit render.
You can certainly use the slider to control the position in real time but you can't slide it about while the system is rendering (or previewing). You could use the slider to control the amount of easing but you would set it and then hit render, You might be able to set the slider in a repeating preview and see the results on the next pass, but it wouldn't be a one to one kind of thing like in your movie.