What happens when the temperature goes below the dew point?
The temperature can never go below the dew point. The dew point is the temperature that 100% relative humidity is reached, based on the amount of water vapour in the air. That means that once the temperature drops to that point, the air cannot hold any more water vapour so condensation occurs. If condensation occurs then the amount of water vapour in the air will drop. If the amount of water vapour in the air drops the dew point will drop.
Looking at the above chart, if we have 20 grams of water vapour in1 kilogram of air at 30 degrees, we are at about 75% relative humidity. If the temperature drops 5 degrees we are at 100% relative humidity. That means that the dew point was 25 degrees. At that point condensation will occur. If the temperature drops another 5 degrees (now at 20 C), we will lose about 5 grams of water vapour as liquid water in that kilogram of air. The relative humidity stays at 100% but the dew point will have dropped.
Long story short, the dew point can equal the dry bulb, but at 100% relative humidity if the dry bulb drops so does the dew point. The highest the dew point can ever be is the dry bulb temperature.