The choice of cleaning the lid on your travel mug will depend on the type of travel mug you have and what material the lid is made from.
Plastic lids are easy to clean and can be popped into the dishwasher. Just make sure if your travel mug has a silicon seal that it won’t come off in the dishwasher. If placing your 360 degree lid into the dishwasher, make sure the button is pushed and you give it a good shake when taking it out of the dishwasher.
I also recommend scrubbing your lid out with hot soapy water after each use. For a thorough clean, you can make up a bicarbonate/vinegar type paste with hot boiling water and pour this into a deep bowl. Place your lid inside and leave for several minutes before rinsing off with boiling water and allowing to dry (preferably overnight). If you have a 360 degree lid, make sure the button is pushed down to get into those nooks and crannies and let it dry out completely with the top of the lid facing down.
For a leakproof travel mug with a choice of lids, the Moon Travel Mug can be purchased here.