First thing:
Create an array of images to store the images, rather than create several almost identical sections of code...
See From Several Variables to Arrays (Processing Wiki).
Then, instead of having... all of that code (quite frankly, it is difficult to look at), you can simply reference the image to draw with a simple formula between the brackets.

Then, (I haven't looked at your code, but it's a typical error), you probably have the animation working correctly, but are displaying it too rapidly. Again, in your formula, simply divide the output by a larger number. Right now, it looks like you're using 10, try 100... It's all about experimentation.

Or you could use this...

1. PImage[] Frames;
2. 
   
3. class Bird {
4.   float x, y;
5.   int whichFrame, frameDelay, timeNextFrame;
6.   Bird( float _x, float _y, int _frameDelay ){
7.     x = _x;
8.     y = _y;
9.     whichFrame = 0;
10.     frameDelay = _frameDelay;
11.     timeNextFrame = millis() + frameDelay;
12.   }
13.   void fly(){
14.     simulate();
15.     render();
16.   }
17.   void simulate(){
18.     if( millis() > timeNextFrame ){
19.       timeNextFrame = millis() + frameDelay;
20.       whichFrame++;
21.       whichFrame%=Frames.length;
22.     }
23.   }
24.   void render(){
25.     image(Frames[whichFrame], x, y); 
26.   }
27. }
28. 
    
29. Bird birdA;
30. Bird birdB;
31. 
    
32. void setup() {
33.   size(400, 400);
34.   background(255);
35.   Frames = new PImage[4];
36.   Frames[0] = loadImage("bird1.png");
37.   Frames[1] = loadImage("bird2.png");
38.   Frames[2] = loadImage("bird3.png");
39.   Frames[3] = loadImage("bird4.png");
40.   birdA = new Bird( random(10,100), random(10,100), 3500 );
41.   birdB = new Bird( random(10,100), random(10,100), 1000 );
42. }
43. 
    
44. void draw() {
45.   background(255);
46.   birdA.fly();
47.   birdB.fly();
48. }

UNTESTED CODE!!!

I put the images into an array (now called Frames), and wrote a Bird class that keeps track of each bird's position, which frame it's on, and how fast it moves to the next frame...

Its position is determined by the member variables x and y. You should be able to write a movement() member function inside the Bird class (it will update some member variables much like the simulate() function does), that will update that bird's position in each frame.

I'd try:

1. void movement(){
2.  x = map( abs((millis()%5000)-2500), 0, 2500, 10, width-10 );
3. }