The unity has a free Cinemachine plugin, which is designed to use the camera to track an object, change plans, and all those chips that are used in the movie. The plugin is a bit complicated to understand, but if you get it right, it will help you make very decent camera spans or make a smart camera to monitor objects. Here are some good video tutorials https://unity3d.com/en/learn/tutorials/topics/animation/using-cinemachine-getting-started
In your case, two cameras are configured, one from the first person, the second from the third. And at the right moment, their significance changes. A special CinemachineBrain controller automatically without your participation will not only make a smooth switching between cameras on the priority that has been set, but will also allow you to configure in the editor how this transition should look.
The plugin has built-in tolerances when the object is moving to avoid jerking when playing animations and this allows you to do a smooth tracking of the object. The camera can either follow the object according to the specified rules, or watch it being in the same place. In general, the plugin is very rich in features, such as observing groups of objects, simulating camera shake when shooting from hands, switching to a camera with a better view, moving the camera along a predetermined path, etc.