Hi Marilyn,
Runny eyes may or may not be a sign of a more serious problem. Oftentimes, I find pets in a new home getting exposed to new things so the potential for some allergies always enters my mind. A simpler explanation might be that as they explore they could get small dust particles in their eyes from reaching those difficult to clean places. When it comes to a warm environment or a heated one, I tend to find the lack of humidity and dryness lead to a natural drying of the mucous membranes, which can include the conjunctiva of the eyes and the lower airways. Medically speaking though, other possibilities could include a clogged nasolacrimal tear duct, conjunctivitis, or even an ingrown hair repeatedly making contact with the cornea. For those reasons, and because they only get 2 eyes, I would recommend getting it looked at by your veterinarian as soon as possible.
I hope this helps.
Dr. Ryan Llera