diff --git a/proxy_components/mock-engine-store/src/mock_store/mock_core.rs b/proxy_components/mock-engine-store/src/mock_store/mock_core.rs
index 268efa55034..90b3af719ee 100644
--- a/proxy_components/mock-engine-store/src/mock_store/mock_core.rs
+++ b/proxy_components/mock-engine-store/src/mock_store/mock_core.rs
@@ -140,11 +140,13 @@ pub fn copy_data_from(
     // kv data in memory
     for cf in 0..3 {
         for (k, v) in &source.data[cf] {
-            if k.as_slice() >= source.region.get_start_key()
-                && k.as_slice() < source.region.get_end_key()
+            if (k.as_slice() >= source.region.get_start_key()
+                || source.region.get_start_key().is_empty())
+                && (k.as_slice() < source.region.get_end_key()
+                    || source.region.get_end_key().is_empty())
             {
                 debug!(
-                    "copy_data_from region {} {:?} {:?} S {:?} E {:?}",
+                    "copy_data_from write to region {} {:?} {:?} S {:?} E {:?}",
                     region_id,
                     k,
                     v,
@@ -152,6 +154,15 @@ pub fn copy_data_from(
                     source.region.get_end_key()
                 );
                 write_kv_in_mem(target, cf, k.as_slice(), v.as_slice());
+            } else {
+                debug!(
+                    "copy_data_from skip write to region {} {:?} {:?} S {:?} E {:?}",
+                    region_id,
+                    k,
+                    v,
+                    source.region.get_start_key(),
+                    source.region.get_end_key()
+                );
             }
         }
     }
diff --git a/proxy_tests/proxy/shared/fast_add_peer.rs b/proxy_tests/proxy/shared/fast_add_peer.rs
index 8cecb5464d1..3138b5c8a48 100644
--- a/proxy_tests/proxy/shared/fast_add_peer.rs
+++ b/proxy_tests/proxy/shared/fast_add_peer.rs
@@ -5,6 +5,7 @@ use crate::utils::v1::*;
 enum SourceType {
     Leader,
     Learner,
+    // The learner coesn't catch up with Leader.
     DelayedLearner,
     InvalidSource,
 }
@@ -328,6 +329,9 @@ fn test_overlap_apply_legacy_in_the_middle() {
     cluster.shutdown();
 }
 
+// `block_wait`: whether we block wait in a MsgAppend handling, or return with
+// WaitForData. `pause`: pause in some core procedures.
+// `check_timeout`: mock and check if FAP timeouts.
 fn simple_fast_add_peer(
     source_type: SourceType,
     block_wait: bool,
@@ -421,7 +425,7 @@ fn simple_fast_add_peer(
         _ => (),
     }
 
-    // Add peer 3
+    // Add peer 3 by FAP
     pd_client.must_add_peer(1, new_learner_peer(3, 3));
     cluster.must_put(b"k2", b"v2");
 
@@ -649,6 +653,7 @@ mod simple_blocked_nopause {}
 mod simple_blocked_pause {
     use super::*;
     // Delay when fetch and build data
+
     #[test]
     fn test_simpleb_from_learner_paused_build() {
         fail::cfg("fap_core_no_fallback", "panic").unwrap();