Does transferring perfume from its spray bottle into a small glass bottle affect its composition?

You might lose some to evaporation etc in the process, but otherwise no.

Throw (place) in your suitcase (checked luggage)?

Yes, it would affect the strength of fragrance. That is why perfumes are stored in air tight bottles (even the non aerosol ones).

In addition to what @AK said, there might be something in the bottle you are pouring into that mixes poorly with the fragrance. You might try a little and see what happens.

^^Soaped, rinsed and dried the bottle twice. Will definitely try this and hoping my nose is good enough to find out how much fragrance was lost.