About message history on linked devices
Web and Desktop
Right after you link a device, your primary phone sends an end-to-end encrypted copy of your most recent message history to your newly linked device, where it’s stored locally. It can take a few minutes for your message history to appear on linked devices depending on the number of messages in your chats.
Note: Not all messages and chats are synced to linked devices from your phone. WhatsApp Desktop syncs more message history than WhatsApp Web. To see or search your full history, check your phone.