Blending Modes Lost When Placing Illustrator File as Smart Object in Photoshop

Blending modes work in any layered file work by specifying the interacting with the layer(s) below. That interaction can happen within the Smart Object, so you can have as many objects and layers within the smart object all with their own blend modes and they will interact as specified with each other. What they cannot do is use those same blend modes to interact with document layer(s) outside of the smart object. A smart object is effectively a container that presents itself within the overall document as a single pixel layer at the pixel size of the main document. As such it can have only a single blend mode for that overall smart object layer.  

You may need to revise the way you bring separately blended elements across, or ensure that the interaction happens solely within the smart object, for example by having a background beneath the blended elements (which can of course be masked where it is not required)

Dave