It is difficult to diagnose the problem with the first mugo pine without a few more pictures or more information. The problem could be related to insects or it could be environmental. Does this bush receive enough water? Is it planted near a sidewalk or street where it could be splashed with de-icing salts in the winter? Has there been any recent digging in the area? And/or do you see any evidence of insects? Look closely at the bark to see if you have any evidence of insects like spots of popcorn-like tan pitch oozing from the bark. If possible, could you send a few more pictures, one picturing the whole shrub and its surroundings and another picture closer up? 

It sounds like the second shrub is experiencing a normal fall needle drop. If the needles that turn brown are on the interior of the branches and the tips are still green, this is a normal function. Evergreens tend to lose their inner needles just like a deciduous tree loses its leaves in the fall.

I do not see any evidence of insects. Here are some additional photos of the plant.  It is a well-established plant.  I have been giving it extra water to see if it made a difference. 

I do not see any evidence of insects. Here are some additional photos of the plant.  It is a well-established plant.  I have been giving it extra water to see if it made a difference. 

If you don't see any evidence of insects the problem is probably environmental; i.e., under or over watering. Mugo pines have a fair drought tolerance, but weeks of hot, dry weather with drying winds like we had last summer can suck the moisture out plants and they can have a difficult time recovering. Also over watering can be just as much of a problem. If the plant gets too much water and the roots are not allowed to dry out somewhat between waterings, that can also cause the die back you are seeing. When you dig down below the surface, the soil should be slightly moist but never too wet. Our clay soils along the front range tend to hold onto the water longer. And if your soil is sandy, the problem may be that the water is draining to rapidly and the roots are unable to take up enough moisture.

For more information about environmental disorders of woody plants, see the link below. The sections on drought injury and over watering may provide some useful information for you.

http://extension.colostate.edu/topic-areas/yard-garden/environmental-disorders-of-woody-plants-2-932...