How to handle illustrator documents with transparent background and blured objects?

Hello, i often buyadobe stock illustrator files with transparent backgrounds (for example AdobeStock_237738943), but i am quite unable to use them properly in indesign or photoshop. Most of the time i get the chequered background when exporting the image – and no trasparency. Things get even worse when for example objects are used which contain blurs or gradients. These appear also not correct. I tried to copy the illustrator document into photoshop using the new layer feature but unfortunately there where too many layers. So that didn't work either. So: how do i handle such documents?1