diff --git a/core/modules/update/src/Tests/UpdateTestBase.php b/core/modules/update/src/Tests/UpdateTestBase.php
index fc39d1a3cc..99312208a9 100644
--- a/core/modules/update/src/Tests/UpdateTestBase.php
+++ b/core/modules/update/src/Tests/UpdateTestBase.php
@@ -2,6 +2,8 @@
 
 namespace Drupal\update\Tests;
 
+@trigger_error(__NAMESPACE__ . '\UpdateTestBase is deprecated in Drupal 8.4.0 and will be removed before Drupal 9.0.0. Instead, use \Drupal\Tests\update\Functional\UpdateTestBase', E_USER_DEPRECATED);
+
 use Drupal\Core\DrupalKernel;
 use Drupal\Core\Url;
 use Drupal\simpletest\WebTestBase;
@@ -22,6 +24,9 @@
  * (via the 'update_test_xml_map' variable), and then performs a series of
  * assertions that the report matches our expectations given the specific
  * initial state and availability scenario.
+ *
+ * @deprecated Scheduled for removal in Drupal 9.0.0.
+ *   Use \Drupal\Tests\update\Functional\UpdateTestBase instead.
  */
 abstract class UpdateTestBase extends WebTestBase {
 
diff --git a/core/modules/update/src/Tests/FileTransferAuthorizeFormTest.php b/core/modules/update/tests/src/Functional/FileTransferAuthorizeFormTest.php
similarity index 92%
rename from core/modules/update/src/Tests/FileTransferAuthorizeFormTest.php
rename to core/modules/update/tests/src/Functional/FileTransferAuthorizeFormTest.php
index 9d1819fca3..d768b06d81 100644
--- a/core/modules/update/src/Tests/FileTransferAuthorizeFormTest.php
+++ b/core/modules/update/tests/src/Functional/FileTransferAuthorizeFormTest.php
@@ -1,6 +1,6 @@
 <?php
 
-namespace Drupal\update\Tests;
+namespace Drupal\Tests\update\Functional;
 
 /**
  * Tests the Update Manager module upload via authorize.php functionality.
@@ -23,7 +23,7 @@ protected function setUp() {
 
     // Create a local cache so the module is not downloaded from drupal.org.
     $cache_directory = _update_manager_cache_directory(TRUE);
-    $validArchiveFile = __DIR__ . '/../../tests/update_test_new_module/8.x-1.0/update_test_new_module.tar.gz';
+    $validArchiveFile = __DIR__ . '/../../update_test_new_module/8.x-1.0/update_test_new_module.tar.gz';
     copy($validArchiveFile, $cache_directory . '/update_test_new_module.tar.gz');
   }
 
diff --git a/core/modules/update/src/Tests/UpdateContribTest.php b/core/modules/update/tests/src/Functional/UpdateContribTest.php
similarity index 99%
rename from core/modules/update/src/Tests/UpdateContribTest.php
rename to core/modules/update/tests/src/Functional/UpdateContribTest.php
index 81f7117a3a..20d397cd48 100644
--- a/core/modules/update/src/Tests/UpdateContribTest.php
+++ b/core/modules/update/tests/src/Functional/UpdateContribTest.php
@@ -1,6 +1,6 @@
 <?php
 
-namespace Drupal\update\Tests;
+namespace Drupal\Tests\update\Functional;
 
 use Drupal\Core\Url;
 use Drupal\Core\Utility\ProjectInfo;
diff --git a/core/modules/update/src/Tests/UpdateCoreTest.php b/core/modules/update/tests/src/Functional/UpdateCoreTest.php
similarity index 98%
rename from core/modules/update/src/Tests/UpdateCoreTest.php
rename to core/modules/update/tests/src/Functional/UpdateCoreTest.php
index b6287c7c04..9a66f65b16 100644
--- a/core/modules/update/src/Tests/UpdateCoreTest.php
+++ b/core/modules/update/tests/src/Functional/UpdateCoreTest.php
@@ -1,8 +1,9 @@
 <?php
 
-namespace Drupal\update\Tests;
+namespace Drupal\Tests\update\Functional;
 
 use Drupal\Core\Url;
+use Drupal\Tests\Traits\Core\CronRunTrait;
 
 /**
  * Tests the Update Manager module through a series of functional tests using
@@ -12,6 +13,8 @@
  */
 class UpdateCoreTest extends UpdateTestBase {
 
+  use CronRunTrait;
+
   /**
    * Modules to enable.
    *
@@ -76,6 +79,7 @@ public function testNormalUpdateAvailable() {
         $this->standardTests();
         $this->drupalGet('admin/reports/updates');
         $this->clickLink(t('Check manually'));
+        $this->drupalGet('admin/reports/updates');
         $this->assertNoText(t('Security update required!'));
         $this->assertRaw(\Drupal::l("8.$minor_version.1" . $extra_version, Url::fromUri("http://example.com/drupal-8-$minor_version-1$extra_version-release")), 'Link to release appears.');
         $this->assertRaw(\Drupal::l(t('Download'), Url::fromUri("http://example.com/drupal-8-$minor_version-1$extra_version.tar.gz")), 'Link to download appears.');
@@ -139,6 +143,7 @@ public function testMajorUpdateAvailable() {
           $this->standardTests();
           $this->drupalGet('admin/reports/updates');
           $this->clickLink(t('Check manually'));
+          $this->drupalGet('admin/reports/updates');
           $this->assertNoText(t('Security update required!'));
           $this->assertRaw(\Drupal::l('9.0.0', Url::fromUri("http://example.com/drupal-9-0-0-release")), 'Link to release appears.');
           $this->assertRaw(\Drupal::l(t('Download'), Url::fromUri("http://example.com/drupal-9-0-0.tar.gz")), 'Link to download appears.');
diff --git a/core/modules/update/src/Tests/UpdateDeleteFileIfStaleTest.php b/core/modules/update/tests/src/Functional/UpdateDeleteFileIfStaleTest.php
similarity index 96%
rename from core/modules/update/src/Tests/UpdateDeleteFileIfStaleTest.php
rename to core/modules/update/tests/src/Functional/UpdateDeleteFileIfStaleTest.php
index 99499d77c2..601a6cc2ff 100644
--- a/core/modules/update/src/Tests/UpdateDeleteFileIfStaleTest.php
+++ b/core/modules/update/tests/src/Functional/UpdateDeleteFileIfStaleTest.php
@@ -1,6 +1,6 @@
 <?php
 
-namespace Drupal\update\Tests;
+namespace Drupal\Tests\update\Functional;
 
 /**
  * Tests the update_delete_file_if_stale() function.
diff --git a/core/modules/update/src/Tests/UpdateTestBase.php b/core/modules/update/tests/src/Functional/UpdateTestBase.php
similarity index 92%
copy from core/modules/update/src/Tests/UpdateTestBase.php
copy to core/modules/update/tests/src/Functional/UpdateTestBase.php
index fc39d1a3cc..9379fe2ab7 100644
--- a/core/modules/update/src/Tests/UpdateTestBase.php
+++ b/core/modules/update/tests/src/Functional/UpdateTestBase.php
@@ -1,10 +1,10 @@
 <?php
 
-namespace Drupal\update\Tests;
+namespace Drupal\Tests\update\Functional;
 
 use Drupal\Core\DrupalKernel;
 use Drupal\Core\Url;
-use Drupal\simpletest\WebTestBase;
+use Drupal\Tests\BrowserTestBase;
 
 /**
  * Defines some shared functions used by all update tests.
@@ -23,7 +23,7 @@
  * assertions that the report matches our expectations given the specific
  * initial state and availability scenario.
  */
-abstract class UpdateTestBase extends WebTestBase {
+abstract class UpdateTestBase extends BrowserTestBase {
 
   protected function setUp() {
     parent::setUp();
@@ -58,7 +58,7 @@ protected function setUp() {
    *   (optional) A string containing the URL to fetch update data from.
    *   Defaults to 'update-test'.
    *
-   * @see Drupal\update_test\Controller\UpdateTestController::updateTest()
+   * @see \Drupal\update_test\Controller\UpdateTestController::updateTest()
    */
   protected function refreshUpdateStatus($xml_map, $url = 'update-test') {
     // Tell the Update Manager module to fetch from the URL provided by
@@ -69,6 +69,7 @@ protected function refreshUpdateStatus($xml_map, $url = 'update-test') {
     // Manually check the update status.
     $this->drupalGet('admin/reports/updates');
     $this->clickLink(t('Check manually'));
+    $this->drupalGet('admin/reports/updates');
   }
 
   /**
diff --git a/core/modules/update/src/Tests/UpdateUploadTest.php b/core/modules/update/tests/src/Functional/UpdateUploadTest.php
similarity index 96%
rename from core/modules/update/src/Tests/UpdateUploadTest.php
rename to core/modules/update/tests/src/Functional/UpdateUploadTest.php
index 80d3f15d29..6096620062 100644
--- a/core/modules/update/src/Tests/UpdateUploadTest.php
+++ b/core/modules/update/tests/src/Functional/UpdateUploadTest.php
@@ -1,10 +1,11 @@
 <?php
 
-namespace Drupal\update\Tests;
+namespace Drupal\Tests\update\Functional;
 
 use Drupal\Core\Extension\InfoParserDynamic;
 use Drupal\Core\Updater\Updater;
 use Drupal\Core\Url;
+use Drupal\Tests\TestFileCreationTrait;
 
 /**
  * Tests the Update Manager module's upload and extraction functionality.
@@ -13,6 +14,10 @@
  */
 class UpdateUploadTest extends UpdateTestBase {
 
+  use TestFileCreationTrait {
+    getTestFiles as drupalGetTestFiles;
+  }
+
   /**
    * Modules to enable.
    *
@@ -50,7 +55,7 @@ public function testUploadModule() {
     // Check to ensure an existing module can't be reinstalled. Also checks that
     // the archive was extracted since we can't know if the module is already
     // installed until after extraction.
-    $validArchiveFile = __DIR__ . '/../../tests/aaa_update_test.tar.gz';
+    $validArchiveFile = __DIR__ . '/../../aaa_update_test.tar.gz';
     $edit = [
       'files[project_upload]' => $validArchiveFile,
     ];
@@ -63,7 +68,7 @@ public function testUploadModule() {
     $moduleUpdater = $updaters['module']['class'];
     $installedInfoFilePath = $this->container->get('update.root') . '/' . $moduleUpdater::getRootDirectoryRelativePath() . '/update_test_new_module/update_test_new_module.info.yml';
     $this->assertFalse(file_exists($installedInfoFilePath), 'The new module does not exist in the filesystem before it is installed with the Update Manager.');
-    $validArchiveFile = __DIR__ . '/../../tests/update_test_new_module/8.x-1.0/update_test_new_module.tar.gz';
+    $validArchiveFile = __DIR__ . '/../../update_test_new_module/8.x-1.0/update_test_new_module.tar.gz';
     $edit = [
       'files[project_upload]' => $validArchiveFile,
     ];