Interpolate a derivative of a solution between subdomain

Hi All,
I have a question regarding Subdomains in FEniCS. I want to use the Gradient of the solution in one Subdomain as a Neumann Boundary Condition for the other Subdomain. I used the command interpolate to exchange the information between the Functionspaces of the Subdomains, which worked fine for the solution itself, but not its Gradient. When is use interpolate for the Gradient i get only the value Zero for the Neumann Boundary Condition, which cant be the case. Is there a guide on how to exchange the gradient between funtionspaces of two subdomains, because i wasnt able to find one on my own. Thank you all very much in advance for reading my question and for looking into it.

Please provide a minimal working example, as it is hard to give any guidance without some concrete code (it would also be very time-consuming for people to make a code based on your description)