Then why does the tattoo fade over time?
It is debated whether all the ink particles are in fibroblasts, or if some remain as extracellular aggregations of ink. Also, the lifespan of the ink containing fibroblasts is not known. Presumably, ink particles are moved into the deeper dermis over time due to the action of mobile phagocytic cells (think immune cells), causing the tattoo to look bluish, faded and blurry. Examination of older tattoos (e.g. 40 years) show that the ink is in the deep dermis, and also found in local lymph nodes. Since some types of phagocytic immune cells migrate to lymph nodes to "present their goods", the discovery of ink in lymph nodes is consistent with the theory of phagocytic cells being the cause of ink movement.
What about the sun?
Sun exposure equals sun damage, whether you realize it or not. Langerhans cells, a type of dendritic cell, are present throughout the epidermis, but mostly located in the stratum spinosum. During sun exposure, many Langerhans cells will undergo apoptosis (a type of cell death where the cell breaks apart into many small fragments) while others migrate into the dermis and a minor inflammatory reaction occurs. The inflammatory reaction is not restricted to the epidermis, but also involves the dermis. Such a reaction causes the recruitment of more phagocytic immune cells to the area.
With the presence of larger than normal amounts of migrating phagocytic cells, the chances of ink movement increases, thus accelerating the fading of the tattoo.