During developing, add more time to the development, e.g. 10min for Tri-X400 pushed to 1600 for Tmax dev.

You are talking here about pushing on film development.

Definition of pushing on Wikipedia:

Push processing in photography, sometimes called uprating, refers to a film developing technique that increases the effective sensitivity of the film being processed.2 Push processing involves developing the film for more time, possibly in combination with a higher temperature, than the manufacturer's recommendations. This technique results in effective overdevelopment of the film, compensating for underexposure in the camera.

Source: Guide to film photography

When pushing or pulling film at different EI speeds, you must over- or under-develop the film to compensate for over or underexposing the film. Generally, pushing the film to another ISO makes the development process much easier. If you push a 400 speed film to EI 800, you should develop the film based on the development time for ISO 800 film. This will give you a solid starting point as you learn to push or pull process film.

This technique applies to both colour and black&white films.

Talking B&W now: As you (probably) already know, the longer you keep the film in the developer, the more the silver halides on the films are reduced to metallic silver which will then get fixed on the film during the fixing process. That means, the more you keep the film in the developer, the more "light" will be caught on film, the more the exposure will be pulled during the development.

You should nevertheless always read the specifications of the developer.

Quoting again from Guide to film photography:

Chemical developers have also provided some general rules for push and pull film processing. Kodak recommends that when push processing, you should increase the development time by two minutes for each camera stop of underexposure. With pull processing, the development time should be decreased one minute for each stop of overexposure. Ilford recommends increasing development time by 20% to boost contrast for underexposed images.

Both companies provide basic data sheets for push/pull processing times similar to their sheets for standard processing times. Please consult the appropriate company for more information.