Think of it this way: TWH (almost always) have the genetics that allow them to do a non-trotting gait just like a TB, QH or WB have the genetics that allow them to trot. When you see a trotting breed at liberty you may see variety of different trots (extension, collection, jogging, etc) whereas you will see many TWH exhibit a running walk, flat walk, pace, trot, rack, etc when out loose. With training you are taking the natural ability of the horse and focusing on what gait you actually want. The quality of the gait in any horse is a combination of ability, training and current circumstances.
For example, the mare I have shown for several years is pretty set in her gait and requires very little assistance to maintain it. As a matter of fact, she is as good or better wearing a bosal in a water glass class, with my attention focused on anything but her, than she is in a formal English class. Now although my 3 year old can do a very nice flat walk and running walk, she requires a lot more attention and use of the aids to keep in the proper frame.
So the short answer to your question is that a TWH usually has ability to do a running walk, the quality of that gait can be enhanced (or will deteriorate) depending on how they are trained and who rides them. Just like any other horse.
For reference: a flat walk and running walk have the same sequence of footfalls, the flat walk is a slower, less extended version of the running walk. In the flat walk, the head should nod and the rear legs exhibit overstride, it is definitely faster than a normal horse walk.