...although it is viable to copy and paste between the adobe suite applications in many cases, care does need to be taken when performing such a function...
...for importing into indesign the recommended workflow for file integrity is to import the native .ai file via file > place...
...you need to save from illustrator as native .ai format, ensuring you have PDF compatibility enabled in the save as dialog (on by default)...
...you also need to ensure your illustrator document raster effects settings are appropriate for the intended output with the correct resolution etc (effects > document raster effects settings). This controls transparent effects produced inside illustrator...
...ensure your illustrator document color mode is set to the intended output color mode, CMYK or RGB (file > document color mode)...
...also ensure your artwork is on the correct size artboard, anything that extends the artboard will not show in indesign, it will be clipped off at the artboard edges...
andrew