I’ve several flutes, all different.
My lightest flute is a maple traverso (“Baroque flute”). It weighs almost nothing. The only issue with this flute is that it is so light that it feels like it may fly out of your hands. Like you could breath some helium, play the flute, and watch the flute float to the ceiling. 
My Hamilton 6-key is also a very light flute, weighing (subjectively) a bit less than the keyless Seery. (Note, all my weights are subjective, I’ve not actually weighed any of these flutes).
The heaviest is without a doubt the M&E 6-key, but although it’s a bit heavy, I’ve never found it very hard to handle. My good friend and student Alec has that flute right now, and just plays the hell out of it. 
My old German 8-key has a lined head. Overall, it’s a moderately heavy flute, not quite as heavy as the M&E, but it is very off-balance towards the head. I find that when I hold this flute I tilt it down towards the flute more than the others but I still use the same three-points-of-contact approach. Picking up this flute is a little awkward because the head is so heavy, but once it’s in playing position holding it there is not an issue.
My Boehm-system silver flute is moderately heavy but very well balanced. I’ve played this flute for so long it feels like a natural extension of my body. I am so used to the peculiarities of this one particular Boehm-system flute that when I try someone else’s flute, it feels very strange to me, though if the flute is in good repair I will be able to play it.
The only change necessary for the silver flute is that the right thumb rests a bit higher on the side of the flute than on the wooden flute, right under the keywork. This doesn’t have anything to do with the weight of the flute, though, it’s a function of getting my right hand pinky in a position to control the touches for the E-flat, C-sharp, C, and B keys.
–James