I'm sure you will have a great hunt here in NZ and the 270 Win is a great cartridge for these animals.

As with any animal, a shot through the shoulder or high into the brisket if the animal is facing you, will drop tahr and chamois. If you are looking at shoulder mounts then a heart/lung shot behind the shoulder will suffice. The country for tahr is high and open mountain tops and can be rough and difficult to get around in, same for chamois although chamois can be found from the mountain tops to sea level in many areas.

Usually best to try and anchor these animals on the spot to prevent falls over bluffs etc where there is a chance the nice set of trophy horns and hide are damaged. Chamois have magnificent eyesight so can spot you a mile away. They will generally stay watching until you get closer then will move off. If they do move off they will not always go far so it pays to keep stalking undercover and close in from another angle if possible. Your guide will of course know all this anyway.

If you have to take a running shot, I have found Tahr generally run straighter and with proper lead you can knock them over at reasonable ranges. Chamois are a different animal and the little shits bounce all over the place make it difficult to hit a moving animal.

If you get to observe these animals much, they will amaze you as to their agility in the mountains, they seem as though they are glued to the mountain faces.

Good luck and good hunting.