When bridging to a rollup there's really two distinct transactions taking place,
an initiating bridge transaction is sent on the origin chain
a relaying/execution transaction is sent on the destination chain.
Payment for this relaying transaction is made when you initiate the bridge, that's why you're required to send a little ETH (or the native gas token for that rollup) when initiating bridge transaction. In cases where the rollup has a custom gas token, you need to pay for the relaying or execution transaction with that custom gas token.
Let's consider a few scenarios,
Rollup has a custom gas token (GAS) and you want to bridge USDC
rollup settles to Arbitrum One
the rollup uses GAS as their gas token instead of ETH
you want to bridge 10 USDC from Arbitrum One to the rollup.
You first need to ensure you have a small GAS balance on Arbitrum One. When you initiate the bridge you'll send that 10 USDC + a small amount of GAS to ensure the relaying transaction is executed.
Rollup has a custom gas token (GAS) and you want to bridge GAS
rollup settles to Arbitrum One
the rollup uses GAS as their gas token instead of ETH
you want to bridge 10 GAS from Arbitrum One to the rollup.
You need to ensure you have more than 10 GAS on Arbitrum One. When you initiate the bridge you'll send that 10 GAS + a small amount of GAS to ensure the relaying transaction is executed.
Rollup uses ETH as its gas token and you want to bridge USDC
rollup settles to Arbitrum One
the rollup uses ETH as their gas token
you want to bridge 10 USDC from Arbitrum One to the rollup.
You first need to ensure you have a small ETH balance on Arbitrum One. When you initiate the bridge you'll send that 10 USDC + a small amount of ETH to ensure the relaying transaction is executed.
Rollup uses ETH as its gas token and you want to bridge ETH
rollup settles to Arbitrum One
the rollup uses ETH as their gas token
you want to bridge 1 ETH from Arbitrum One to the rollup.
You first need to ensure you have more than 1 ETH on Arbitrum One. When you initiate the bridge you'll send that 1 ETH + a small amount of ETH to ensure the relaying transaction is executed.