Check out import records, You can use that to import all the data, or just the data in the current found set from one filemaker table (or selected other files) into a different table in filemaker. There are lots of options, so read up on the tool in Filemaker Help and make a back up copy of your file before importing so you can toss the file and try again if you don't get the results you expect.
On a side note, Copy and Paste are usually best left unused when moving data around. Copy destroys any data your user might have previously copied to the clip board and both steps only work if the referenced field is on the current layout. Future edits of a layout can thus break these scripts and you won't know they're broken until you notice that data is not being copied and/or pasted.
For moving smaller amounts of data use either Set FIeld or set field in combination with set variable instead of copy/paste.
You can use the import function instead
You may need to use the match field and update and add new records that are the same if you plan to continue to use both databases.
Or you can create a temporary table in each database add a button to create a record in the temp table for each record you wish to transfer. (you only want to do this once so you need to create a relationship between the tables to check that it isnt already there)
When you want to transfer you import from this temp table into the receiving database.