skin is made of two layers mainly- Epidermis and Dermis.
Epidermis- is the most superficial layer of skin. layer which make the complexion of skin. it consists of multiple layers. from above down- stratum corneum, stratum lucidum, stratum granulosum, stratum spinosum and stratum basale.
Dermis- is the deeper layer of skin below epidermis. this layer contains blood vessels. this is the layer in which dye is injected. after injection body immune cells becomes active to remove this foreign ink matter and engulf the ink. after few days epidermis heals up and the damage layer of epidermis is replaced by new epidermis cells.
so ink now lie below the epidermis in Phagocytic cells in dermis. healing in dermis ultimately forms collagen that trap the ink permanently in dermis.
After decades new layers in epidermis and dermis form so the ink becomes more deeper and deeper. Tattoo becomes degraded. ink can be seen in the draining lymph node of that part of skin. phagocytic cells eat the ink and take it to the lymph nodes.