A roundup of excellent research and guidelines for stretching, warm-up and preventing injury from American College of Sports Medicine, Sport Medicine UK, American Heart Association, and National Center for Biotechnology Information.

From The American College of Sports Medicine -ACSM

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Muscle strains, tendonitis and overuse injuries can be reduced.
Every workout must begin with a warm-up and end with a cool-down. A warm-up is necessary to prepare the body for exercise by increasing heart rate and blood flow to working muscles. The warm-up should start slow and easy and consist of a general cardiovascular exercise such as walking, jogging or biking. The goal is to break a sweat.
 

From Sport Medicine UK

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What is a warm-up?
A warm-up is a session which takes place prior to doing physical activity; usually a warm-up will consist of light cardiovascular exercises combined with stretches.
What happens in a warm-up?
Most warm-up sessions will include a combination of cardiovascular exercises, stretching and strength drills. The cardiovascular exercises are designed to increase circulation, increase body temperature and bring the heart rate up, while stretching warms the muscles and prepares them for the movements they will be required of them during the activity. Explosive strength exercises, which may include sprint drills or jumps, gently increase the level of intensity and prepare the body for sudden movements in the game which will follow; these exercises should only be done once the muscles are warm; this will prevent injuries.
Effects of the warm-up
The warm-up should gently prepare the body for exercises by gradually increasing the heart rate and circulation; this will loosen the joints and increase blood flow to the muscles. Stretching the muscles prepares them for physical activity and prevents injuries. The warm-up is also a good opportunity for an individual to prepare themselves mentally for the game ahead and for a team to work together prior to the start of the game. Warm-ups can also be used to practice skills and team drills.
How long should a warm up last?
Most warm up sessions last between 20 minutes and half an hour; this gives the body plenty of time to gradually get ready for physical activity and gives the player time to prepare themselves mentally.
 

From The American Heart Association

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“Warming up and cooling down are good for your exercise performance — you’ll do better, faster, stronger — and for your heart since the increased work on the heart ‘steps up’ with exercise,” said Richard Stein, M.D., professor of cardiology in the Department of Medicine at New York University and co-director of Cardiology Consult Services.
“Stretching also makes many people feel better during and after exercise and in some people decreases muscle pain and stiffness.” When done properly, stretching activities increase flexibility.
So what’s the big deal?
A good warm-up before a workout dilates your blood vessels, ensuring that your muscles are well supplied with oxygen. It also raises your muscles’ temperature for optimal flexibility and efficiency. By slowly raising your heart rate, the warm-up also helps minimize stress on your heart.“Warming up before any workout or sport is critical for preventing injury and prepping your body,” said Johnny Lee, M.D., director of the Asian Heart Initiative at the New York University Langone Medical Center and president of New York Heart Associates in New York City.
“Stretching allows for greater range of motion and eases the stress on the joints and tendons, which could potentially prevent injury. Warming up, such as low-heart rate cardio, prepares the circulatory and respiratory system for the upcoming ‘age- and type-appropriate target heart rate’ exercising, whether it’s endurance or sprint type of activities.”
The cool-down is just as critical.  It keeps the blood flowing throughout the body. Stopping suddenly can cause lightheadedness because your heart rate and blood pressure drop rapidly.
Warm up
Before you exercise, think about warming up your muscles like you would warm up your car. It increases the temperature and flexibility of your muscles, and helps you be more efficient and safer during your workout.  A warm-up before moderate- or vigorous-intensity aerobic activity allows a gradual increase in heart rate and breathing at the start of the activity.
Tips:

• Warm up for 5 to 10 minutes. The more intense the activity, the longer the warm-up.
• Do whatever activity you plan on doing (running, walking, cycling, etc.) at a slower pace (jog, walk slowly).
• Use your entire body. For many people, walking on a treadmill and doing some modified bent-knee push-ups will suffice.

Stretch duration and quantity — what’s the right dosage

From The National Center for Biotechnology Information

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Stretching Research
Many studies have evaluated various effects of different types and durations of stretching. Outcomes of these studies can be categorized as either acute or training effects. Acute effects measure the immediate results of stretching, while training effects are the results of stretching over a period of time. Stretching studies also vary by the different muscles or muscle groups that are being examined and the variety of populations studied, thereby making interpretation and recommendations somewhat difficult and relative. Each of these factors must therefore be considered when making conclusions based on research studies. Several systematic reviews of stretching are available to provide general recommendations.
Recommendations
Static, dynamic, and pre-contraction stretching are all effective methods of increasing flexibility and muscle extensibility; however, these modes may be more effective in specific populations. Several authors have noted an individualized response to stretching;(48,56,60) therefore, stretching programs may need to be individualized.
 

From American College of Sports Medicine

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How much physical activity is needed to improve health and cardio-respiratory fitness?
What volume of physical activity is needed? Several studies have supported a dose–response relationship between chronic physical activity levels and health outcomes (155,372), such that greater benefit is associated with higher amounts of physical activity. Data regarding the specific quantity and quality of physical activity for the attainment of the health benefits are less clear. Epidemiologic studies have estimated the volume of physical activity needed to achieve specific health benefits, typically expressed as kilocalories per week (kcalIwkj1 ), MET-minutes per week (METIminIwkj1 ), or MET-hours per week (METIhIwkj1 ). Large prospective cohort studies of diverse populations (216,237,320,353) clearly show that an energy expenditure of approximately 1000 kcalIwkj1 of moderate-intensity physical activity (or about 150 minIwkj1 ) is associated with lower rates of CVD and premature mortality. This is equivalent to an intensity of about 3–5.9 METs (for people weighing È68–91 kg) and 10 METIhIwkj1 . Ten MET-hours per week can also be achieved with Q20 minIdj1 of vigorous-intensity (QÈ6 METs) physical activity performed Q3 dIwkj1 or for a total of È75 minIwkj1 . Previous investigations have suggested that there may be a dose–response relationship between energy expenditure and depression, but additional study is needed to confirm this possibility (25,101). In the general population, this 1000 kcalIwkj1 volume of physical activity is accumulated through a combination of physical activities and exercise of varying intensities. Therefore, the 2008 Physical Activity Guidelines for Americans (370), the 2007 AHA/ACSM recommendations (155,264), and the ACSM guidelines (14) allow for a combination of moderate- and vigorous-intensity activities to expend the requisite weekly energy expenditure. An intriguing observation from several studies is that significant risk reductions for CVD disease and premature mortality begin to be observed at volumes below these recommended targets, starting at about one-half of the recommended volume (i.e., È500 kcalIwkj1 )
 

Other Resources

https://gmb.io/stretching/
http://yourlivingbody.com/2015/01/05/how-long-to-hold-a-stretch/
Do Stretch Durations Affect Muscle Mechanical and Neurophysiological Properties?
Effect of acute static stretch on maximal muscle performance: a systematic review
Differential effects of 30- vs. 60-second static muscle stretching on vertical jump performance
 
Move. Heal. RESEARCH. Evolve