diff --git a/live-build/functions b/live-build/functions
index 8a6777f2..d9ae727b 100644
--- a/live-build/functions
+++ b/live-build/functions
@@ -158,10 +158,17 @@ teardown_mountpoint() {
 
     # 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
+
+    # Make all submounts private
     for submount in $(awk </proc/self/mounts "\$2 ~ /$mountpoint_match/ \
                       { print \$2 }" | LC_ALL=C sort -r); do
         mount --make-private $submount
+    done
+
+    # Now we can unmount them
+    # 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 $submount
     done
     recover_sourceslist "${mountpoint}"