Gas Welding heats the Cast Iron slower than Arc based processes and the flame is lower temperature than an arc.  This means Carbon migration is not normally a problem.  Use of a proprietary Cast iron Gas Welding rod is important.  The only real downsides to Gas Welding Cast Iron is the amount of heat needed if components are large.  Gas Welding is also a slow process.

Arc/Stick Welding is, arguably, the best all-round process for Welding Cast Iron, provided the correct welding rods are used.  Cast iron Welding Rods have a special Graphite rich flux, this graphite chemically ties up the Carbon in the Cast Iron, limiting migration into the weld metal and heat affected zone.  There are two common types of Cast iron Welding Rod, Ferro-Nickel and Pure Nickel.  Ferro-Nickel are typically 53% Steel and 47% Nickel.  Ferro-Nickel Rods are cheaper than pure Nickel and are ideal for welding Cast Iron to Steel.  Pure Nickel will produce a softer, more malleable weld deposit.  I would advocate using Ferro-Nickel, unless the job specifically requires Pure Nickel.

Mig Welding is, in my opinion, not a great way to weld Cast Iron.  Whilst there are specialist Flux Cored Wires available, unless you have a repeating application that you can create a procedure for, I would not advocate Mig Welding Cast iron.

Tig Welding is not considered a suitable process for Welding Cast Iron.  An open arc process such as Tig offers no opportunity to mitigate Carbon migration.