Induction cooking works by generating an electric current in the metal cooking vessel and converting that current into heat, which requires a resistive material (i.e. a poor conductor).
It's a bit of a catch-22, because you need a good conductor to actually distribute that heat. This is why some of the best induction cookware is clad metal - two layers of (magnetic) steel around an inner layer of highly-conductive aluminum or sometimes copper, sometimes layered multiple times this way. The thick magnetic sheets generate heat, and the thin conductive sheets transfer it.
Thicker metal means that the surface is slightly farther away from the magnetic field source, but also has significantly more resistance (since resistance is proportional to length, and we are talking about doubling or even tripling that), so theoretically it should be much more effective at converting the electric current to heat, and heat up faster.
The disadvantage of a thicker metal - assuming that there is no additional heat transfer material - is hot spots. This doesn't matter at all for a skillet, but if you're trying to cook with, say, a cast-iron stock pot or dutch oven, you'll find that the hot spots are even worse on an induction cooktop than an electric or gas cooktop, since the surface heat is generated very rapidly but takes forever to spread.
Carbon steel is a pretty good compromise, which is why it's generally the material of choice for woks, and that's the material I'd choose for an induction cooktop for any vessel other than a skillet or frying pan (assuming I had to choose a single material; again, copper/aluminum-clad stainless steel works better). For pans/skillets, you want the heat to stay focused on the surface, which makes heavy/thick cast iron a better choice.
I can't honestly say that I've had much experience with forged iron, but all my instincts are telling me not to bother with it because it would be combining the worst of both worlds - lower resistance at the surface but still relatively high resistance up the sides. The only caveat to this would be a possibly significant difference between the magnetic susceptibility/permeability of forged vs. cast iron, but this is likely going to vary from vessel to vessel anyway (not all cast/forged iron is the same) so I think your best bet there is to try sticking a magnet to it. If it sticks much harder to the forged than cast iron (doubtful) then it might be good for surface cooking; I still wouldn't choose it for a larger pot.
If you get an enameled cast iron pot too hot, you can soften the enamel enough to damage it. My brother ruined one when we were younger on an electric stove, set on high (was making ramen, forgot about it, turned off the stove after the water had boiled off, when it cooled it fused to the burner)
The thing about camp fires is that they burn at different temperatures depending on the fuel and how much air's getting into it.
As you're using a dutch oven, I assume that you're doing something that requires low and slow cooking ... so you should be okay. If I were doing this, I'd try to pay attention to the fire (put the palm of your hand near the coals -- if you can't keep it there for a few seconds, it's too hot; move it somewhere else). And make sure that you have enough liquid in there that it won't risk going dry. If necessary, you can make a flour & water dough to use to seal the lip of the pan shut, so steam can't escape.
I'd also soap the bottom of the pot -- take a liquid dish soap, and rub the bottom and lower edges of the outside, so that when soot starts to deposit, it's depositing on the layer of soap, and not the pot itself. (it makes cleaning so much easier).
I'd also beware of grills for another reason -- many manufacturers are making lighter weight grill parts to save materials (so it costs them less to make). Many are so thin that I doubt they'd take the weight of a heavy cast iron pot. (I've broken one using a cast iron griddle)
It's also worth mentioning that what you have isn't quite ideal for campfire cooking, as it doesn't have the little legs to stand if off the ground a little bit, or the lip on top so that you can put coals on top, but be able to lift the lid without the ashes falling in your food. You can get around problem #1 with the grate, and might able to do something about #2 with some creative use of aluminum foil.
Best Answer
Staub has cast iron enameled outside and bare iron inside