Make the object's new location somewhere in between the old location and the current mouse location.

Easing determines the movement speed: none, instantly, something in between. Example:

1. float x,y;
2. float oldx, oldy;
3. float easing = 0.05;
4. 
   
5. void setup() {
6.   size(500,500);
7.   smooth();
8.   noStroke();
9.   fill(0);
10. }
11. 
    
12. void draw() {
13.   background(255);
14. 
    
15.   x = (1-easing) * oldx + easing * mouseX;
16.   y = (1-easing) * oldy + easing * mouseY;
17. 
    
18.   ellipse(x,y,50,50);
19.  
20.   oldx = x;
21.   oldy = y;
22. }