It would be unusual to use beef or lamb stock in a chicken gravy. But it's not unusual to add chicken stock to a beef or lamb gravy. Veal is another popular option, famed for its richness. As with any gravy, the quality of the stock accounts for much of the flavour, so try to use a good homemade stock.
It's common to add red wine to the gravy along with the stock, and a dash of port adds a wonderful richness. Redcurrant jelly is a traditional sweetener – with one or two teaspoons often being added to lamb gravy. Other popular additions are tomato paste, Vegimite or Worcestershire sauce.