There's really no just one answer to fit them all.
PNG is an uncompressed file format and most (if not all) video formats are somewhat compressed. That's where some colour info is lost or simply changed/adjusted to the given/available colours for each specific codec, format...
In order to preserve this particular shade of blue, you'll need to adjust the output settings. Either through the format or more likely through choosing and adjusting the available codec/s. Moho is not really meant for such tasks (even After Effects can't properly export h.264) hence my recommendation for Img sequence export from Moho and final rendering done in some video editor, where you can adjust all of these things towards your liking.
On the other hand....
As you may see here, I've used your uploaded image and rendered it through the Moho (mp4/h.264 - aac) and the colours remained pretty much the same. This is a screenshot with video player in top half and your image in the bottom half

But, this is a still image video where compression in video rendering is not the same for still and moving images/shapes.
And one more thing, which is quite indicative - note how's the shading on the character's neck rendered without that gradient. The colour space is pretty much narrowed in the mp4 hence the lack of gradient and all of the colours needed for proper display of that main blue.