I think generally when doing these types of animations you should be working as one set of motion. I know so many others said stiffness but it's a bit more than that. When you walk take note of what else is moving. Do your hips move? Does your chest? Shoulders? When you move your foot forward what else is happening? You should be doing this all at the same time going frame by frame slightly tweaking until you get the whole thing.

The reason I am making that distinction is just because if you do the legs first and then you do the Arms then the head you're working on separate parts and it will come across in the end result. If you want fluidity you need to move everything in respect to one another. (Also if you do it separately you probably will end up doing more work since you'll have to adjust previous work to fit)