diff -u b/src/Plugin/Block/MenuBlock.php b/src/Plugin/Block/MenuBlock.php
--- b/src/Plugin/Block/MenuBlock.php
+++ b/src/Plugin/Block/MenuBlock.php
@@ -80,11 +80,2 @@
    */
-  public function getConfiguration() {
-    $label = $this->getBlockLabel() ?: $this->label();
-    $this->setConfigurationValue('label', $label);
-    return $this->configuration;
-  }
-
-  /**
-   * {@inheritdoc}
-   */
   public function blockForm($form, FormStateInterface $form_state) {
@@ -325,9 +316,17 @@
     $tree = $this->menuTree->transform($tree, $manipulators);
     $build = $this->menuTree->build($tree);
 
+    $label = $this->getBlockLabel() ?: $this->label();
+    // Set the block's #title (label) to the dynamic value.
+    $build['#title'] = [
+      '#markup' => $label,
+    ];
     if (!empty($build['#theme'])) {
       // Add the configuration for use in menu_block_theme_suggestions_menu().
       $build['#menu_block_configuration'] = $this->configuration;
+      // Set the generated label into the configuration array so it is
+      // propagated to the theme preprocessor and template(s) as needed.
+      $build['#menu_block_configuration']['label'] = $label;
       // Remove the menu name-based suggestion so we can control its precedence
       // better in menu_block_theme_suggestions_menu().
       $build['#theme'] = 'menu';
only in patch2:
unchanged:
--- a/menu_block.module
+++ b/menu_block.module
@@ -100,3 +100,19 @@ function menu_block_theme_suggestions_menu(array $variables) {
 
   return $suggestions;
 }
+
+/**
+ * Implements hook_preprocess_hook() for "block".
+ *
+ * Set the block label with the #menu_block_configuration label if it exists.
+ */
+function menu_block_preprocess_block(&$variables) {
+  if (isset($variables['content']['#menu_block_configuration']['label'])) {
+    $config_label = $variables['content']['#menu_block_configuration']['label'];
+    // Some block twig templates (especially classy + bartik from core) use
+    // `{{ configuration.label }}` to print the label. Others just use
+    // `{{ label }}`. Therefore, we have to set both template variables for
+    // this to work consistently.
+    $variables['configuration']['label'] = $variables['label'] = $config_label;
+  }
+}