I don't think you have to treat that topic significantly different than on lower altitudes (but I have to admit, I have never been higher than 4300 m) as long as you stick to trekking.
As you already introduce your question, warming up is especially important for sports where maximum strength (e.g. climbing) and/or explosive muscle movement (e.g. sprinting) are required or where parts of your body are moved to the full possible range of mobility (e.g. acrobatics). Other than you, I wouldn't classify trekking into none of these categories. Therefore I would characterize trekking an activity that can require significant physical condition (especially in higher altitudes) but is – if we omit walking downhill with a heavy pack – rather gentle regarding muscles and the skeletal system. As such, I would not say that you have to take very special care about warming up.
There are two things to remember: your blood will get thicker due to height acclimatisation, therefore the blood circulation into your fingers and toes can decrease and the overall temperature is colder than you're used to. The former is basically a thing to keep in mind and react on if you have the impression that it affects your extremities. The latter is mainly a matter of preparation, i.e. right clothing and gear. Above that, cold climate is something one can get used to to some extent.
If you have prepared well, you have brought a sleeping bag and pad that kept you warm and comfy all night, not just a bit warm like in "didn't die from freezing". Waking up warm and not shivering from the cold will already be an important piece of getting ready to go. The rest should be done more or less automatically by your normal morning routine:
If you're tent camping, preparing a breakfast, unpitching your tent, and packing all your stuff should give you enough amount of movement to get your muscles warm, even in cold weather. If you have slept in a lodge, hut, bed and breakfast, or the like (whatever rather simple but solid accommodations are called locally), the ambient inside should be significantly warmer than in a tent. This will compensate for the less movement you will get, as you don't have to pack that much stuff here.
In any case, the key is not to waste much time in the morning as you will get cold if you sit around for too long doing nothing. Then just start to walk rather slowly. Don't rush away from the start but begin with a unhurried pace until your body is fully warm.
If you have had an unpleasant night where you didn't sleep well and feel cold or just out of any reason have the feeling, that you are not quite warm in the morning, you can do the same stuff that you would possibly do at home as well: hop around a bit, do some squats, swing your arms around etc. to increase blood flow and get especially your limbs warm.
The above should be enough as long as you're only trekking. If you are climbing (high intensity movement) you will surely have to warm up more thoroughly. This is especially true if you have to start climbing hard right away from the camp without any approach walk that would get you warm.