How to transfer BTC from desktop wallet to mobile wallet (without fees), say using Electrum or Bitcoin Core I have both Bitcoin Core &amp; Electrum wallets installed on both my desktop &amp; mobile.  How do I transfer BTCs from respective desktop to mobile wallets (&amp; vice-versa) directly (i.e., not thru the blockchain) to avoid fees?
To accomplish this you will not 'transfer the bitcoin', you will just import the keys that can spend your bitcoin, into a new wallet.
Not all wallet software will support importing/exporting keys in every format, so you'll need to ensure that your current wallet software is capable of exporting keys in a format the the new wallet will accept. There are several formats for private keys, as well as other standards for backing up/recreating wallets, such as BIP32/39/44/84, etc. Be sure to familiarize yourself with these formats, and know which ones you'll need to use in order to accomplish your goal. Any wallet software worth using will allow you to export keys in some way or another, and hopefully provide some sort of documentation to help with this.
Very importantly, you need to understand that the private keys (or mnemonic seed phrase) that you'll be handling is the literal key to your bitcoin. If you store it on, or enter it into a compromised device, your bitcoin could be stolen.
Once you have the keys that control your bitcoins exported, you can import them to the new wallet, and (potentially after a rescan of the blockchain to locate your funds), your funds should appear in the new wallet.