To my understanding, the gradient in the video is animated (the colours aren't changing, they're moving). You can achieve this by animating the background-position.

Example (you can of course play with all the settings):

.home-page .hero-section.slide {
  /* Your existing code */

  animation: animationName 10s ease infinite;
}

@keyframes animationName {
  0% { background-position: 0% 50%; }
  50% { background-position: 100% 50%; }
  100% { background-position: 0% 50%; }
}

I personally like using CSS Gradient Animator to help me achieve this, that way I have something to go off of.