Hello Cheryl
( thanks for the note on the system. It should have allowed you to select Michigan as your state, and then displayed a list of Michigan counties)

Euonymus alatus, burning bush, is not tolerant of wet or flooded soils and too wet soil can cause roots to die and shrubs to wilt. 
If bark is damaged at the base or elsewhere along the branch they could be infected with fungi that would cause rot such as in your pictures.
Crown galls can also develop and stop the flow of water along the stems, causing the plant to wilt on those stems.

There are a couple ways to get a diagnosis:
You can email you pictures and information to MSU Plant 1-Diagnostic lab. Your last picture of an injury on a branch was very blurry, so please take that again in good light. Also take pictures of he wilted leaves and of any damage at the base of the shrub. There is no charge for this service, and they will give you an opinion. <personal data hidden>
If they cannot give a reliable opinion they may request that you send in plant samples for diagnosis. There is a fee for analysis of plant samples, and the fee, sample submission form, and instructions are on their website. https://www.canr.msu.edu/pestid/

2-Your second option is to hire a certified arborist to come on site and asses the whole planting. He/she will look at roots, stems, leaves and diagnose issues and recommend a plant care plan, how to prune, etc.

Renovation pruning, rejuvenation pruning, or pruning out rotted, dead and diseased wood may be recommended. But first, you want to locate the source of the rotted wood and wilted stems. Save some of the rotted branches for diagnosis by the lab or an arborist.
https://www.canr.msu.edu/news/spring_shape_up_tidying_up_your_plants

Here are some references for you—
Certified arborists search tool www.treesaregood.org

Common pests and diseases of burning bush:
insect pests of burning bush site:edu
https://entomology.ca.uky.edu/ef428

https://hgic.clemson.edu/factsheet/euonymus-diseases-insect-pests/

Rots of woody plants- there are several and the lab or an arborist can identify them-
http://ipm.ucanr.edu/PMG/PESTNOTES/pn74109.html

Thank you Laura. You have given me a lot of options. I have looked before for an arborist and it was very hard to find one. I will pursue a different option.

Hello Cheryl
Pictures are always helpful but cannot always eliminate all possible causes for rot setting in. Most rots can be pruned out, so that is why I gave you the pruning article. 
Determining where/why damage occurs helps prevent it in future( for example, if you regularly are getting vole damage at the base, wrapping the base with a trunk guard or hardware cloth every fall-and removing it in spring -will help prevent rabbits and voles from gnawing  the trunk.

I tried searching on one of the zip codes for Wayne county and the result came up with many certified arborists. Here is the list the ISA Arborist site gave me for 48202. You can go there and set it for your zip code(http://www.treesaregood.org)
https://www.treesaregood.org/findanarborist/findanarborist

Try the MSU Lab which staffed by  plant pathologists, and they may be able to tell you more from just the pictures. It is free via email, so definitely worth trying.

Burning bush are no longer recommended because they self seed in our natural areas. However, controlled well in the landscape, they are beautiful.
Best regards, Laura

Thank you again and thank you for the arborist help. Very helpful.

Yes, burning bush are considered hardy here.( I have one that came with my house, and I have been here 30 years)

Any damage to the trunk can be harmful, because it opens it to disease and insects. The living part of the wood on a tree or shrub lies just below the bark and is only a few millimeters thick. If an animal is hungry in winter, they often chew bark for food, rabbits and voles do this often. If they chew all the way around a  stem, that is called ‘girdling the stem’. Water can no longer move from the roots up across the chewed area, and the stem above dies.

Burning bush produces tiny flowers which mature into seeds. Birds and other wildlife can carry these to other locations. When the shrubs begin to grow in our natural areas, they are so vigorous they force out native plants; this is why the are no longer recommended. They can become “invasive” in natural areas like forests, along rivers.
http://www.misin.msu.edu/facts/detail/?project=NA&id=63&cname=Winged%20burningbush

I am glad you find the information helpful! Good luck with your shrubs.

I hope to be as wise as you are some day!