Bridges among EVM compatible SideChains (L1): Problems

I've been experimenting with bridges such as AnySwap, Celer and HopExchange to move assets across different EVM compatible chains, but with all of them I'm facing the same issue:

• Most of the bridges don't allow to send their native asset from blockchain_a to blockchain_b

This results in having a frozen asset that can't be moved. For example, if I move ETH from Avalanche to BSC, then I can't do anything with these ETH in BSC, because I need the BNB token to pay for the fees.

I haven't found any bridge that allows me to send BNB over any bridge to BSC. So I end up with funds that I can't move (assuming of course that I don't use a binance CEX account to fund the address)

Have anyone ran into the same issue?