Baby is in an upright cradle carry:

Reach into the sling with your non sling arm.  Lift the baby up under their arm, while pushing up on his/her bottom on the outside of the sling with your sling arm.  Pushing up on the baby’s bottom releases tension on the fabric and makes it easier to lift the baby out.  Leaning forward slightly helps also.

Continue pushing up on the baby’s bottom with your sling arm and lift the baby up on to your chest—Once the baby’s upper body is free of the sling, you can lean back slightly to use your chest/shoulder to help support the baby’s weight.

and up to your shoulder–

and up into the traditional “burp” position:

Once baby is up high on your shoulder, you can use your non sling arm to hold the baby there, and use your sling arm to pull the fabric down free of the baby’s legs:

Baby is now out of the sling: