Even though ARDL has put to rest the debate of using I(0) and I(1) variable within same model, but with the introduction of new variables a new debate has came forward that how to incorporate I(2) variables in the same set.
Yesterday I found one reference online (thanks to Hossain Academy Facebook, shared by Mosikari Teboho) the picture blow is the discussion which states first the definition of cointegration. That is if two series are of order I(1) then they are only cointegrated if their resultant residuals are having lower order i.e. I(0).
This intuition is utilized by the Harris (1995) from where the above source is taken. According to the author if you have I(2), and I(1) variables in the model and the dependent variable is I(1) then you can isolate the I(2) variables and make an equation such that the residuals formed from it are I(1). Use that residuals as independent variable against the I(1) dependent variable to form residuals which are of I(0) order. This way you can measure the effect of I(2) variable through the I(1) residuals to I(1) dependent variable.
This model can be further extended like take the I(0) residuals from second equation to put it against I(0) dependent variable.
If we have original dependent variable I(1) then we can use I(0) variables in second equation and form ARDL cointegrating bounds like mentioned here, here and here. But there is one limitation to this idea that what if our original dependent is I(2), I would suggest that in such case take first difference of I(2) variable and use it as I(1) variable as dependent.
Harris, R. I. (1995). Using cointegration analysis in econometric modelling (Vol. 82). London: Prentice Hall.
Pagan, A. R., & Wickens, M. R. (1989). A survey of some recent econometric methods. The Economic Journal, 962-1025.