Each airport tends to be a little different on how they handle runway crossings. We try our best to emulate how it's done at the various ZLA airports (to the extent we can).
How would this play out at a larger airport, e.g. Altanta? Let's say you're parked at the "South Cargo Ramp" and are departing runway 26R, i.e. you've got 3 runways to cross. Surely you wouldn't be calling TOWER (and which Tower?!) for each of those?
Well, I happen to fly in and out of ATL on a regular basis (3 times today alone actually), so here it how this would work here:
First off runway 26R isn't used for departures, runway 26L is. However to keep in the spirit of the question we will say you are re-positioning from South Cargo to North Cargo.
I'll start by saying that this is a long event (20-30 minutes), so it is quite insane. By the time we are done we will have used 7 different ATC frequencies (we will assume it's somewhat busy)
The aircraft would start by calling Ground on 121.65 for initial taxi. The most likely instruction here is "Taxi via R12, R, hold short runway 27L"
Approaching 27L "monitor tower 119.3"
On 119.3
"Cross 27L, turn left on N, hold short Papa"
Approaching P (yes on the other end of the runway)
"At Papa, cross 27R, join L, contact ground 121.75" (Note departures from 27R depart from intersection M2, so there is no chance of a head to head here)
Once on L you'll notice there isn't actually a way to get to the North side of the airport without either using a ramp, or the non movement area of taxiway Dixie (named Dixie to prevent confusion with Delta airlines, which has a couple departures a day from ATL). Ground will advise what ramp you'll go through (most likely D), so...
"Taxi via L, Dixie, contact ramp tower 131.37"
Even though it's a non movement area, it's still controlled by a ramp tower (run by Delta airlines, or the city of Atlanta).
Ramp will provide the transition North via Dixie, and tell you to "hold short F, contact ground 121.90"
This new ground control will tell you to "taxi via Dixie, hold short runway 26L, monitor tower 125.32"
At some point this tower will tell you to "Cross 26L, hold short runway 26R, monitor tower, 119.10"
Soon followed by "Cross 26R, join A, contact ground 121.90 (again)"
Ground will taxi you to the North cargo via A.
Fun Fact: During the big ATL hub pushes there are 19 controllers working in the ATL tower, or the ramp towers (directly moving aircraft or assisting in the moving of aircraft). 5 Towers, 3 ground controllers, 2 metering positions, Clearance Delivery, and 8 ramp towers).
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So what's the name of the game here, listen to ATC and they will guide you where you need to be and tell you who to contact. If you're in doubt ask (as every airport is different, very different, trust me on that one).