Ace
Without knowing the exact type of cast iron the old iron barrels are made from I can't say what the coef of thermal expansion is but here's a few data points:
Gray Cast Iron per ASTM A-48 = 6.0 X 10e-6 in/in/deg F.
" " " per ASTM A-436 = 6.6 X 10e-6 in/in/deg F.
Ductile or nodular cast irons = 6.6 X 10e-6 in/in/deg F.
Malleable cast iron = 7.5 X 10e-6 in/in/deg F.
The only wrought steel I have any data on is 4340.
It's coef of thermal expansion at 400 degrees is 7.4 X 10e-6 in/in/deg F.
Taking the worst case situation between these values shows a difference of 7.4-6.0=1.4 X10e-6 in/in/deg F. so with a bore of 87mm (3.425 in) the expected bore expansion at 400 degrees F would be .049mm (.0019) less in cast iron.
Needless to say, if you have more reliable numbers for the materials involved use them.
If you don't, be sure to check my math before making any chips.
While I was typing all of this, you posted saying you are using a iron sleeve.
If it is malleable iron, notice that its coef of therm expansion is about the same as the 4340.