These procedures require using the Terminal app, in your Applications/Utilities folder.

Be extremely careful when using Terminal.  It is a direct command-line interface into UNIX, the underpinning of OSX.  Unlike the Finder, there are few protections against making a mistake, which can cause untold damage.

In Terminal, the prompt looks like this:   <ComputerName>:~  <your name>$

(where <your name> is your short user name).  It's followed by a non-blinking block cursor.  When you first start it, you'll see something like this:

      

The commands below are prefaced by the term "sudo" which means to execute the command with "root" or "super user" privileges.  The first time you use it in a session, you'll see some warnings and a prompt for your Admin password (nothing will be displayed as you type your password).  Press Return again.

Normally, there are few error or confirmation messages in Terminal.   When a command completes, you usually just see another prompt.

"Inherit" a backup

Copy the following after the prompt, and leave a space, but do not press Return yet:

                            sudo tmutil inheritbackup

If your backups are on an external HD, locate and open the drive in the Finder window.  At the top level of the drive is a Backups.backupdb folder containing a folder named for your old Mac, per the sample.  This is what you want the new Mac to "inherit." Drag that folder to the Terminal window. 

(If your backups are on a Time Capsule or other network location, find the sparse bundle containing the backups for your old Mac via the Finder, per the green box below, and drag it to the Terminal window):

                                   

Terminal will supply the "path" to the dragged folder or sparse bundle (underlined in red):

                            

Note that it will insert a back slash (\) before any spaces in the names.  That's normal.

Press Return, enter your Admin password if prompted (it won't be shown) and press Return again.

"Associate" an OSX volume

To "associate" the OSX volume you're running from, copy the following after the prompt, leave a space after the slash, but do not press Return yet:

                            sudo tmutil associatedisk -a /

Be sure there's a space after the slash.  Open the backup drive in the Finder window (if your backups are on a network, see the green box below).  At the top level of the drive is a Backups.backupdb folder containing a folder named for your old Mac.  Inside that are a series of time-stamped folders.  Inside each of those is one folder for each volume (disk or partition) that was backed-up.  Locate any folder for the "old" volume, but it's usually best to use the latest backup that you're sure was made from the "old" drive).   Drag it to the Terminal window:

Terminal will supply the "path" to the dragged folder (underlined in blue):  

         

Note that it will insert a back slash (\) before any spaces in the names.  That's normal.

Press Return, enter your Admin password, if prompted (it won't be shown) and press Return again.

Note that Time Machine will change the name of the disk on the existing backups to the name of the new one.  In the sample, if the new OSX disk is named Lion, the disk name on all the old backups would be changed from Macintosh HD to Lion.  (That may take a few moments.)

"Associate" a data-only volume (or a second OSX volume)

This is similar to the procedure in the pink box above.  There, the first slash represents the current OSX volume, but in this case, you want the new data-only or second OSX volume.

Copy the following after the prompt, leave a space after the -a, but do not press Return yet:

                            sudo tmutil associatedisk -a 

From a Finder window, drag the new disk/partition to the Terminal window.  It will supply the path as above.  If it's not followed by a space, add one, but do not press Return yet.

Now follow the instructions in the pink box to drag the old volume from the Finder window containing the backups to the Terminal window, and press Return.  The final command should look something like this (the new data-only drive is "NewDataDrive", underlined in orange;  and the backups of the old one were on "M.Misc", underlined in blue):

Note that it will insert a back slash (\) before any spaces in the names.  That's normal.

As in the pink box, if the disk names are different, Time Machine will change the old name on the backups to the new one.  (That may take a few moments.)

Before starting:   Log on as an Admin user, and make sure the backups in question are connected to your Mac.

Open a Finder window to the backups, preferably in List or Column view, and leave it on the screen, as you'll be dragging things from it into the Terminal window as detailed below. 

If your backups are on a network

Follow the instructions in the blue and/or pink box above as desired, to the instructions to use this box to get the network backups in a Finder window:

1. 1.In a Finder sidebar (samples in
   

2. Column View), click the network
   

3. server (Time Capsule or NAS)
   

4. name. 
   

5.                 
   

6. If the server isn't shown, from a
   

7. Finder menubar, select Preferences
   

8. > Sidebar and make sure the
   

9. Connected Servers box is checked.
   

2. 2.Find the disk/volume, connect to
   

3. it if necessary, and double-click it. 
   

4. That should show the sparse bundle
   

5. named for your Mac.  (The name
   

6. may not include ".sparsebundle.")
   

3. 3.To "inherit" the backup, return to
   

4. the blue box above and drag the
   

5. sparse bundle into the tmutil
   

6. command.  Do not mount or open
   

7. the sparse bundle -- if it's already
   

8. mounted, eject it.
   

4. 4.To "associate" the backups to a new or erased drive, double-click the sparse bundle to mount it.  After a moment, the Time Machine Backups (or Backups of <computer>) disk image should appear in the sidebar (green circle).
   

5.                       
   

6. If it doesn't, from a Finder menubar, select Preferences > Sidebar and
   

7. make sure the External Disks box is checked.
   

5. 5.Double-click that to open it.  That will show another Finder window, showing the Backups.backupdb folder as in the sample in the blue box above.
   

6. 6.Scroll down.  If there's an ".inProgress" package, and you've had several failed backups, you may want to delete it.  Connect via Ethernet cable if at all possible -- it may take a long time, but be 2-3 times faster than WIFI.  You'll have to enter your Admin Password.  Hold the Alt/Option key while emptying the trash -- that may take a long time.
   

7. 7.Follow the rest of the instructions in the pink box.
   

8. 8.Eject the Time Machine Backups disk image.
   

9. 9.Open the Activity Monitor app, in your Applications/Utilities folder.  Select All processes in the toolbar, then sort by Process Name (click in the header). If there's a backupd process listed, select it and click the Quit Process icon in the toolbar.  Select Quit in the prompt.
   

10. 10.Whew!  Now try running a backup.  It will take a while (so connect via an Ethernet cable if at all possible).