What affects the cost
Most blockchain transactions require a native network asset to pay gas. If you hold only the token you want to bridge, the wallet may still be unable to submit the transaction.
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Why You Need Gas to Bridge Crypto.
Why a token balance alone may not be enough to complete a cross-chain transfer.
Why the quote changes
ERC-20 transfers can also require an approval transaction before the bridge can spend the selected token. That approval itself can require gas.
What to compare
The exact requirements depend on the network and route. A live quote and wallet prompt will show the transactions that need to be signed.
Practical takeaway
Keep a reasonable native gas balance on the source network and avoid sending it away before completing the bridge. After arrival, check the destination gas requirements for your next transaction.