Mixing different batteries, whether in parallel or in series, is a bad idea. Even enough of the same type of battery wired together is a bad idea.
Batteries are complicated electro-chemical devices. They vary quite a bit with temperature, age, state of charge, discharge rate, and what your dog had for breakfast. There is enough variation between "identical" batteries coming off the same assembly line that you have to be careful in combining them. With different batteries, the answer is simply "don't do that".
With primary cells, there is more forgiveness since they are used once until dead, and therefore future ability to be rechargeable and hold a predictable amount of energy is irrelevant. As such, it is allowable to let primary cells possibly suffer some damage as you try to get the last bit of energy from them. However, if you go too far, they could rupture and cause physical damage to whatever equipment they are contained in.
Rechargeable cells have to be treated much more carefully. Even just putting 4 cells, for example, in series is not so simple. One cell will inevitably have lower capacity than the others. This cell could be discharged so far as to cause damage, even when the stack voltage looks OK (the average cell voltage is above the damage limit).
Charging has the same issue. The lowest capacity cell can be overcharged and damaged before the highest capacity cell is really full.
Properly designed systems with multiple rechargeable cells in series will have monitoring of individual cells, and usually some kind of "charge balancing" circuitry. This shunts some of the charge current around the higher voltage cells to let the lower voltage cells catch up. On discharge, the current is stopped as soon as the lowest cell gets to the point where continuing would cause damage.
To make things more tricky, there isn't a hard line between damage and no damage. There are gray areas where deeper discharge or higher charge starts to decrease lifetime of the cell. That lifetime is itself really a probability.
It gets complicated, and proper multi-cell battery management is a deep topic and will get significant design attention in real professional systems intended for a long life.
Added in response to comment
The kanine nurishment dependency can have particularly tricky influence on performance, so has attracted much research. According to a study commissioned by the Project for Object Oriented Paradigms and performed at the Boise Institute for Technical Excellence, this complex relationship can be distilled down to one simple to understand graph:
Yes, you can charge your batteries with the 5 volts.
1) You would need a boost converter to boost the 5 volts up to the voltage needed to charge the batteries, which would eventually be 8.4 volts.
2) You will need a constant current/constant voltage power charger so that the batteries are charged correctly,
3) Lithium batteries in series need to balanced if they are charged in series. This means that you need to add a battery management chip to your charger that will balance the voltage of the batteries, and make sure that the batteries are not over charged or under charged - very important. Lithium ion will catch on fire or worse if not managed properly.
4) It might be easier to have a boost converter and ONE lithium ion battery and just boost the voltage of that battery to what your watch needs. It would make the charging of the battery easier. Of course you would need to find a battery of sufficient capacity.
Good luck
Best Answer
If you connect those batteries in series, you'll get the sum of the rated voltages: 1.24 + 1.31 + 1.3 + 1.27 = 5.1VDC but this is true on unloaded state.
When the batteries are connected in series, the capacity will be practically the minimum one of the rated capacities, not the sum. Since you claim that each battery has a capacity of 2000mAh, total capacity will be 2000mAh when they are connected in series.
Practically, it can be said that the lifetime will be approximately 2000mAh / 500mA = 4 hours, but please consider the discharge curves. Like the one below:
According to the graph, one battery voltage will be about 1.25VDC for a 500mA load. Then the total voltage will be 1.25 x 4 = 5VDC. If the load current remains constant then the lifetime will be about 4 hours.