Mentioning of the GBA games is funny. I'm a huge crybaby and I didn't find them at all moving. A few tense moments and things like Lyon's love for Eirika kind of pulled me a little but that's about it.
Now, bring on Tellius. I have cried many times with those games, Path of Radiance especially. Hate her all you want, I felt for Mist when Greil took his fall. I cried with Elincia as she felt helpless as a queen. The constant plot twists and surprises in Radiant Dawn were actually unpredictable. The Tellius games to me, as a huge critic of writing, were the pinnacle of the series. They explored new heights and revitalized storytelling under the name of the franchise.
FE1-4... is good for its time. For Thracia 776, I'll need a refresher to give a good opinion.
Edit - To further reiterate.
I have never, ever been more attached to any group in Fire Emblem, than the Greil Mercenaries. A few of their personalities have been seen many times before, but then there are crown jewels like Soren. Soren goes for the throat. Holds nothing back. When Nasir asked him if he was suspected of being a Daein spy, he just responds with a flat out "Yes.". No sugar coating here. Just a small sample of his brilliance.
The GBA games for me were predictable and the least entertaining. I'm never playing FE7 again. I find only characters like Innes say things that both sound realistic and standout. A real personality and a very defined vibe.