In Streets of Rage 4, the final battle against the Y twins or Ms and Mr Y starts immediately after your fight with Ms. Y. In the first phase of the battle they use the movements that you already familiar with. Mr. Y starts firing 2 times and throwing grenades. This first round of attack is relatively easier to escape.
One of the tactics you can use when you see that he is going to shoot, you must position yourself at a different height than him and reach him from that elevation, because this time he only fires in a straight line. After his two shots, he will launch the rocket, so it is better to be close to him here.
When he is going to throw the grenades, he pauses a bit, so you have time to walk away and wait for them to detonate. Ms Y. continues her routine of attacks. If you see her blink red, she's going to try to grab you. Wait for her to get close to you and, just before, jump and hit her with an aerial blow. On the other hand, if she picks up the sword and gives a white flash, she'll try to make you a combo attack. At that time it is better to walk away.
From that point on, always stay on the far left of the screen, best in the upper left corner. The routine is maintained with Ms. Y, who will now be closer to you, so it is easier to perform more combos to her. Take advantage when the leftmost mech tentacle comes to rest in the upper left corner. Give him all the combos you can and you will gradually decrease his health. Keep this routine until the end, alternating blows to Ms. Y and the mech and you will win.