Check option A

$Fe \xrightarrow{dil.\,H_2SO_4} FeSO_4$ This is correct: $Fe + H_2SO_4(dil.) \rightarrow FeSO_4 + H_2$

Next: $FeSO_4 \xrightarrow{H_2SO_4,\,O_2} Fe_2(SO_4)_3$ Ferrous sulfate can be oxidized to ferric sulfate in acidic medium, so this step is acceptable.

Last step: $Fe_2(SO_4)_3 \xrightarrow{heat} Fe$ This is wrong. On heating, ferric sulfate does not directly give iron metal. It decomposes to ferric oxide and sulfur oxides: $Fe_2(SO_4)_3 \xrightarrow{\Delta} Fe_2O_3 + 3SO_3$ Hence, A is incorrect.

Check option B

$Fe \xrightarrow{O_2,\,heat} FeO$ This is not the usual direct oxidation product of iron in air/oxygen. Iron generally forms: $Fe_3O_4 \text{ or } Fe_2O_3$ not pure $FeO$ directly.

Also: $FeSO_4 \xrightarrow{heat} Fe$ is again wrong. On heating: $2FeSO_4 \xrightarrow{\Delta} Fe_2O_3 + SO_2 + SO_3$ So B is incorrect.

Check option C

$Fe \xrightarrow{Cl_2,\,heat} FeCl_3$ This is correct.

Next: $FeCl_3 \xrightarrow{heat,\,air} FeCl_2$ This is wrong. Air would favor oxidation, not reduction. $FeCl_3$ does not convert to $FeCl_2$ by heating in air.

Though: $FeCl_2 \xrightarrow{Zn} Fe$ is possible since zinc is more reactive, the middle step is invalid.

Hence, C is incorrect.

Check option D

$Fe \xrightarrow{O_2,\,heat} Fe_3O_4$ This is correct. Iron on heating in oxygen forms magnetite: $3Fe + 2O_2 \rightarrow Fe_3O_4$

Next: $Fe_3O_4 \xrightarrow{CO,\,600^\circ C} FeO$ This is consistent with stepwise reduction of iron oxides by CO.

Then: $FeO \xrightarrow{CO,\,700^\circ C} Fe$ This is also correct: $FeO + CO \rightarrow Fe + CO_2$

Thus, D is correct.

Final conclusion

The correctly represented series is: $\boxed{D}$