Custom filed link multiple control document

can we link a single custom filed with control document for example I create the custom filed for purchase line can I populate same custom filed data for my sales lines plus I same data I want to populate in inventory form too

You can have custom fields with the same labels on multiple forms. But entering data in one will not populate the others.

However, if you have identically named custom fields on two transaction forms and copy one form to another, the contents of the custom field on the source form will carry over to the destination form.

This only works for form-level custom fields. It does not work for line-item custom fields. It also will not carry any content to a custom field for an inventory item.