diff --git a/ams/AssetMigrator.cs b/ams/AssetMigrator.cs
index f802f4c..7101148 100644
--- a/ams/AssetMigrator.cs
+++ b/ams/AssetMigrator.cs
@@ -212,7 +212,7 @@ public async Task<MigrationResult> MigrateAsync(
                                     result.Status = transformResult.Status;
                                     result.OutputPath = transformResult.OutputPath;
 
-                                    if (result.Status == MigrationStatus.Failed)
+                                    if (result.Status != MigrationStatus.Skipped)
                                     {
                                         break;
                                     }
diff --git a/ams/StorageMigrator.cs b/ams/StorageMigrator.cs
index b1502c4..72586b9 100644
--- a/ams/StorageMigrator.cs
+++ b/ams/StorageMigrator.cs
@@ -176,7 +176,7 @@ private async Task<MigrationResult> MigrateAsync(
                             result.Status = transformResult.Status;
                             result.OutputPath = transformResult.OutputPath;
 
-                            if (result.Status == MigrationStatus.Failed)
+                            if (result.Status != MigrationStatus.Skipped)
                             {
                                 break;
                             }
diff --git a/transform/PackageTransform.cs b/transform/PackageTransform.cs
index 60292ce..7bc74ca 100644
--- a/transform/PackageTransform.cs
+++ b/transform/PackageTransform.cs
@@ -150,7 +150,10 @@ await Task.WhenAll(
                 Directory.Delete(workingDirectory, true);
             }
 
-            return $"{outputPath.Prefix}{Path.GetFileNameWithoutExtension(manifest.FileName)}";
+            // Mark the output container appropriately so that it won't be used as an input asset in new run.
+            await UpdateOutputStatus(outputPath.Container, cancellationToken);
+
+            return outputPath.Prefix;
         }
 
         private UploadPipe CreateUpload(string filePath, UploadHelper helper)
diff --git a/transform/StorageTransform.cs b/transform/StorageTransform.cs
index 4102555..becce1d 100644
--- a/transform/StorageTransform.cs
+++ b/transform/StorageTransform.cs
@@ -42,12 +42,6 @@ public async Task<AssetMigrationResult> RunAsync(
             var result = new AssetMigrationResult();
 
             _logger.LogTrace("Asset {asset} is in format: {format}.", details.AssetName, details.Manifest?.Format);
-            if (details.Manifest != null && details.Manifest.IsLive)
-            {
-                _logger.LogWarning("Skipping asset {asset} which is from a running live event. Rerun the migration after the live event is stopped.", details.AssetName);
-                result.Status = MigrationStatus.Skipped;
-                return result;
-            }
 
             if (IsSupported(details))
             {
@@ -62,6 +56,16 @@ public async Task<AssetMigrationResult> RunAsync(
                     result.Status = MigrationStatus.Failed;
                 }
             }
+            else
+            {
+                var format = details.Manifest != null ? details.Manifest.Format : "non_ism";
+                _logger.LogInformation("The asset {asset} with format {format} is not supported by transform {transform} in current version, try next transform...", 
+                    details.AssetName,
+                    format,
+                    this.GetType().Name);
+
+                result.Status = MigrationStatus.Skipped;
+            }
 
             return result;
         }