Finger and hand swelling is relatively common during running, although sometimes it is very subtle. It probably occurs more frequently when it is hot, but it is not a sign of dehydration and would be unlikely to occur if you were truly dehydrated. Hand swelling can be found in hyponatremia, which occurs in runners who drink too much fluid over a four- to six-hour run.
Related: The dangers of over-hydrating
During exercise, circulation increases and the hand has a large network of small blood vessels that open up. With the increased blood flow, there is some fluid leak between the cells. (We think of blood vessels like a garden hose, but in reality they have links between the cells that allow some leakage, and that leakage increases with the increased circulation from exercise.) This leakage is probably the cause of your finger swelling.
The arm swing of running increases air movement across the skin (to improve heat exchange with the air), and this arm swing motion may also contribute to fluid retention in your hands.
This fluid is eventually reabsorbed into the cells or cleared by the lymph system. This process is going on while you are running, but the rate of removal is slower than the rate of accumulation. Once you stop exercising, the fluid will reabsorb into the vascular system or surrounding cells or be removed by lymph flow.
If you notice that your hand swelling is worse when you run in the heat, you might consider moving your hard or long workouts to the cooler parts of the day. You might also consider taking off any rings before you run, so they do not get trapped on your fingers when they swell.