Die-off designates any reactions of your body being overwhelmed with toxin/dead organism. Skin rashes, diarrhea, kidney pain, bladder infection, etc there are many different die-off symptoms.

The liver has two major functions in your body
1- to produce the digestive enzymes (bile)
2- to clean up the blood and thus clean up your entire body as the toxins will be transported into the blood and then evacuated through the bowels or kidneys (or else evacuated via perspiration).

By keeping a high level of bile in the bowels, the liver also thus ensure that no organism can just walk in. THe bile will kill them if your liver is functioning properly. So in other words, the liver is in charge of bowels die off.

The lymph, otherwise is the space in between the cells, the lymphocytes or white blood cells are in charge of finding any organism there that should not be around and drag it out, through your skin or through the blood.

In fact, die=off is usually a sign that you are going a little too fast for your metabolism and you should slow down (reduce whatever pruduct you are currently using) in order to let your body adjust, it is not a case of lymphs only or liver only it involves many different cleansing aspects of your body.
Die off is actually a sign that your body is not protected enough against the toxins/dead cells. On their way out, they oxydize the cells around them and if your body does not have enough antioxydant (vitamin A, C , E etc) they will cause damage that in turn will cause die -off.