Good breath support can help reduce disfluency in speech. My students and I say that words are like a surfboard and our breath is like the water. The words need to flow out with our breath. When a child blocks or repeats sounds, breath flow tends to be altered.
In order to focus on breathing, we first work on taking breaths from our diaphragm (or belly to the kids).
Place one hand on your stomach. Breathe so that your hand moves out when you inhale, and in when you exhale. This is relaxed breathing. Now switch back to upper-chest typical breathing. Practice switching back and forth to feel the difference.
Practice reading or speaking using belly breathing. You should feel more relaxed. A negative to belly breathing is that it is hard to get more than a phrase or sentence out on one breath. Because of that, students are not encouraged to use this technique all the time, but only prior to and during times when you actively want to relax and facilitate smoother speech.
And as you get better at this technique, you'll find something in between chest and belly breathing. This "in between" breathing will be more relaxed than chest breathing but allow normal conversational pace.
Elongating sounds can help your child slide into words or slide through tough parts in the middle of speech. To practice this, take a reading paragraph and do one of the following.
1) Pick a couple of sounds such as /m/ or /s/. Your child should know which sounds can be stretched and which cannot. Take turns reading sentences and stretching the sound(s) out each time you come to it.
2) Stretch the first sound of each word out if possible. Aim for about two seconds per sound.
3) Keep going until stretched sounds are easily recognized and produced in reading when needed.