If you have a non-standard starter as defined above, and it's healthy, then keep on keepin' on. You're doing fine.
If you have a non-standard starter which is not healthy. Then convert the starter to a standard starter by using the "Sweetening the Pot with 1 Tablespoon" method below to create a standard, white-flour-only starter. Use 1 tablespoon of your nonstandard starter to begin the process. If the starter is not very healthy after a single treatment, then refrigerate the starter for no less than 12 hours, and sweeten the pot again. If the starter is very unhealthy, you may have to repeat the process up to 5 or 6 (or more?) times. Each time you repeat the process, use 1 tablespoon of starter from the last run and discard the rest.
Once you've restored the health of your starter by converting it to a standard starter as described, you may pursue either of 2 methods for converting back to the nonstandard starter that you started with:
B. No Re-Conversion Method:
In the first method, you never really do convert back. Rather, you just maintain your standard starter using standard replenishing techniques as described below. Then when you wish to have that special starter for a particular recipe, then use 1 tablespoon of your standard starter and follow the directions for sweetening the pot, BUT instead of using plain, white flour and plain water, substitute your special flour(s) and liquid(s). For example, a rye starter can be made in one day by taking a single tablespoon of standard starter and mixing it with 1 1/2 cups rye flour and 1 cup water and proofing for 24 hours at 72 to 77 degrees.
C. Re-Conversion Method:
In the second method, you use 1 tablespoon of the newly-refreshed standard starter, then blend it with your special flour(s) and liquid(s), and proof for 24 hours at 72 to 77 degrees. THEN from this time on, continue to feed and replenish your special starter with your special ingredients. If your starter should ever get unhealthy again, then just follow the above procedure to revive it again. Try to determine why your starter is becoming unhealthy. Are you carefully controlling the proofing temperature so the proof is not actually under/over proofing the starter? Underproof prevents the maintenance of high levels of yeast and lacto bacilli in your starter. Overproofing results in yeast and lacto bacilli dying from too much alcohol or acidity in the starter. Are you adding sugar(s) or other simple carbohydrates that cause the microorganisms to grow to fast? Growing to fast would cause the starter to need shortened proofing periods, hence normal proofing periods are actually overproofing the starter. If your starter care passes these tests, then you may consider the possibility that the mix of flour(s) and liquid(s) that you are using are not sufficient for long-term maintenance of your nonstandard starter. In that case, I suggest the first method (above) for maintaining your nonstandard starter.