There are two types of lithium battery to think about here
1] The lithium polymer type, called LiPo in the radio control world. These are small, light, and have a large energy capacity. One significant issue with these is that they must not be discharged too deeply, if they are then they can be damaged. 4QD controllers were not originally designed to work with LiPo but the 4QD-200 / 300 have an adjustable low voltage cut-off, and the DNOs can have a simple modification done to work with them. We are looking at ways to introduce this feature on other controllers.
2] The lithium ion lead acid replacement type [LiFePO4]. These are usually a higher capacity replacement for a standard lead acid battery and very common in the golf buggy world. These usually work ok with 4QD controllers but some batteries have internal protection circuits that can cause unexpected results e.g. over current protection, and low voltage cut-off.