A thorough warm-up has the following physiological benefits:
The way that muscles grow and become stronger is through rebuilding from repeated microtrauma, sustained during exercise. An accumulation of this microtrauma can sometimes lead to small trigger points, which can reduce the efficiency of the muscle. Unless you have a personal myotherapist at your disposal for every single training session or competition (wouldn't that be nice?), an apparatus like a foam roller can be a useful tool for releasing these trigger points. Furthermore, SMR can help to relax muscles before training or competition.
Practical Tip: spend about 30 seconds rolling a muscle group, stopping to apply pressure if you find a sore spot.
Switching muscles on in the warm-up will lead to those muscles being able to fire more quickly during training or competition. During muscles activation or movement preparation, we can target large, powerful muscle groups, or smaller, stabilising muscles that may be otherwise a bit more difficult to wake-up. Some common muscle activation exercises are glute walks (crab steps, monster walks, etc.) with a resistance band around the knees.
Practical Tip: after SMR, spend a few minutes activating smaller, stabilising muscles to prepare them for what's ahead. For lower-body, try glute walk sets; for upper-body, rotator cuff activation may be appropriate.
This is almost an extension of muscle activation. This part of a warm-up involves working through a muscle group's range of motion, while executing sport-specific movement pattern, thus preparing the muscles for movement they'll be expected to perform during the training session or competition. Thus, your main focus should be on muscles that are about to be used. It is important to begin with smaller movement, and progress to movements that require larger ranges of motion. Movements should also go from general to specific in nature, and from low intensity to a higher intensity.
Practical Tip: Spend some time working through the movement patterns required for your sport or training session. Begin with smaller ranges of motion and progress to larger ranges. Begin with general movements and progress to specific movements or skills, and start at a lower intensity, building to a higher intensity towards the end.
There is plenty of evidence to show that static stretching prior to training or competition can be detrimental to muscle function (such as decreased strength, power and reflexes), and thus is not advised as part of a warm-up. Those with muscle length issues should choose to perform static stretching outside of training or competition days.