This is a reasonable thing to do. It violates no “rules” of epidemiology.
But the terminology is important. There are some suggestions about how best to do this. And an important caveat.
In a case-control study, one selects cases (people with a disease) and controls (people without the disease) in order to answer a question about the association of the disease with an exposure. One could legitimately study the same people to answer questions about the association of being diseased or non-diseased and an outcome.
The initial case-control study should—like all good case-control studies--be “population-based.” That is, the cases and the controls need to come from the same defined population—for example, everyone in a state or a city. Ideally, all of the new cases that occur over a fixed period of time in the population would be identified and entered in the study. The controls would be (ideally) a random sample of non-cases during the same period of time. The cases and controls would be characterized by a past exposure.
In the example, 100 people with a new diagnosis of diabetes (let’s make it Type 1 diabetes to be concrete) during a period of 1 year in Los Angeles county (to be concrete) are identified along with 100 controls matched by age and sex and (ideally) chosen at random from all residents of Los Angeles county.
In the case-control study, a set of questions is asked about past exposures.
For example, what is the association of Type 1 diabetes with birthweight? What is the association of Type 1 diabetes with having been breastfed as an infant? What is the association of Type 1 diabetes with having been vaccinated for rotavirus in infancy? Again, in a case-control study all of these questions pertain to an exposure that occurred before the development (or non-development) of Type 1 diabetes.
Once the case-control study is done, these same subjects could be studied longitudinally (going forward in time) to address questions about having the disease and the outcomes of having or not having the disease. The subjects should no longer be called cases and controls to avoid confusion about the design of the longitudinal aspect of the study of these people. One is now conducting a cohort (or an outcomes) study of a systematic sample of people with and without the disease of interest.
In the example, the people with Type 1 diabetes or no Type 1 diabetes could be studied longitudinally to assess the incidence of incidence renal failure or cardiovascular disease or albuminuria in people with and without Type 1 diabetes.
Note, in example, the subjects in the case-control study are matched by age and sex. Any conclusion about the incidence of a disease in people with and without the disease would be generalizable only to people with the same distribution of age and sex. This may or may not be a problem.
Note also that, an individually matched design makes analysis of data more complicated both in the case-control study and in a longitudinal follow-up of the same people. Further, in an individually matched design, if there is non-response in the case-control study or drop-outs in the (attempted) longitudinal follow-up of the same people, there is a loss of statistical power.
The case-control component can be used to assess the association of disease with exposures. The longitudinal component can be used to assess associations of disease with outcome.
CAVEAT. The longitudinal component cannot used to assess the association of exposures with the disease that is the topic of the case-control study. To study exposures and other diseases in the longitudinal component, it is probably best to restrict follow-up to the controls unless there is certainty that the disease that is the topic of the case-control study does not modify the effect of the exposure on other disease or effect modification is of interest.
It is important to get the terminology right. And to do the selection of subjects carefully. And maybe don’t match individually in the case-control study. Rather, select controls so that the distribution of age and sex is (approximately) the same as in the cases.