Since the diagrams are saved in table dtproperties in each database, you could copy the data from one database to another. Here is an example which has been tested:
set identity_insert <targetdb>.dbo.dtproperties on
go
insert into <targetdb>.dbo.dtproperties
(id, objectid, property, value, uvalue, lvalue, version )
SELECT id, objectid, property, value, uvalue, lvalue, version
FROM <sourceDb>.dbo.dtproperties
go
set identity_insert <targetdb>.dbo.dtproperties off
go
SQL = Scarcely Qualifies as a Language