Transfer Related Module / List Data from Lead to Converted Contact

Transfer Related Module / List Data from Lead to Converted Contact

Hi, 

We have a custom related module / list that we use for our Leads. 

Is there any way to make sure that this data is transferred to the corresponding Contact record when the Lead is converted? 

The related list is available to both Leads and Contacts via the appropriate Look Up fields. 

At present the data is lost upon conversion as it only stays associated to the Lead record. 

Ideally this related list / module data would transfer on conversion to a Contact in the same way that Activity, Notes and Attachments data does.