diff --git a/debian/changelog b/debian/changelog
index 3080d6f1..f6282592 100644
--- a/debian/changelog
+++ b/debian/changelog
@@ -1,10 +1,18 @@
-livecd-rootfs (2.408.19) xenial; urgency=medium
+livecd-rootfs (2.408.19) UNRELEASED; urgency=medium
 
   [ Nishanth Aravamudan ]
   * live-build/ubuntu-cpc/hooks/061-open-iscsi.chroot: generate iSCSI
     Initiator Name at first iscsid run for cloud images to ensure it is
     unique (LP: #1444992).
 
+  [ Steve Langasek ]
+  * Improve teardown_mountpoint to recursively find all submounts and
+    unmount them, instead of working from a hard-coded list.  This makes
+    the code resilient against other submounts being added later, including
+    downstream.
+  * Also nuke the sleep / udevadm settle calls in the process, which should
+    never be required and slow down the builds.
+
  -- Victor Tapia <victor.tapia@canonical.com>  Tue, 19 Sep 2017 10:55:11 +0200
 
 livecd-rootfs (2.408.18) xenial; urgency=medium
diff --git a/live-build/ubuntu-cpc/functions b/live-build/ubuntu-cpc/functions
index 3e9e904f..01cbfc92 100644
--- a/live-build/ubuntu-cpc/functions
+++ b/live-build/ubuntu-cpc/functions
@@ -98,16 +98,13 @@ teardown_mountpoint() {
     # Reverse the operations from setup_mountpoint
     local mountpoint="$1"
 
-    umount "$mountpoint/var/cache/apt"
-    umount "$mountpoint/var/lib/apt"
-    umount "$mountpoint/tmp"
-    umount "$mountpoint/sys"
-    umount "$mountpoint/proc"
-    umount -R "$mountpoint/dev"
-
-    apt-get install -qqy udev
-    udevadm settle
-    sleep 3
+    # ensure we have exactly one trailing slash, and escape all slashes for awk
+    mountpoint_match=$(echo "$mountpoint" | sed -e's,/$,,; s,/,\\/,g;')'\/'
+    # sort -r ensures that deeper mountpoints are unmounted first
+    for submount in $(awk </proc/self/mounts "\$2 ~ /$mountpoint_match/ \
+                      { print \$2 }" | LC_ALL=C sort -r); do
+        umount $submount
+    done
     mv resolv.conf.tmp "$mountpoint/etc/resolv.conf"
 }
 
diff --git a/live-build/ubuntu-cpc/hooks/033-disk-image-uefi.binary b/live-build/ubuntu-cpc/hooks/033-disk-image-uefi.binary
index 81782b4b..dadadb16 100755
--- a/live-build/ubuntu-cpc/hooks/033-disk-image-uefi.binary
+++ b/live-build/ubuntu-cpc/hooks/033-disk-image-uefi.binary
@@ -119,11 +119,8 @@ EOF
     chroot mountpoint apt-get -y clean
 
     rm mountpoint/tmp/device.map
-    sync
     umount mountpoint/boot/efi
-    sleep 5
     apt-get install -qqy udev
-    udevadm settle
     mount
     umount_partition mountpoint
     rmdir mountpoint