diff --git a/docs/setup/setting-up-versioning.md b/docs/setup/setting-up-versioning.md
index 8ebd42783c6..cc90634fa47 100644
--- a/docs/setup/setting-up-versioning.md
+++ b/docs/setup/setting-up-versioning.md
@@ -64,7 +64,6 @@ MkDocs implements this behavior by default, but there are a few caveats:
 - the [`site_url`][mkdocs.site_url] must be set correctly in `mkdocs.yml`.
   See the ["Publishing a new version"](#publishing-a-new-version) section for
   an example.
-- you must be viewing the site at that URL (and not locally, for example).
 - the redirect happens via JavaScript and there is no way to know which page you
   will be redirected to ahead of time.
 
diff --git a/src/templates/assets/javascripts/integrations/version/findurl/index.ts b/src/templates/assets/javascripts/integrations/version/findurl/index.ts
new file mode 100644
index 00000000000..717ed1ec483
--- /dev/null
+++ b/src/templates/assets/javascripts/integrations/version/findurl/index.ts
@@ -0,0 +1,135 @@
+import { Sitemap } from "../../sitemap"
+
+/** See docstring for `selectedVersionCorrespondingURL` for the meaning of these fields. */
+type CorrespondingURLParams = {
+  selectedVersionSitemap: Sitemap
+  selectedVersionBaseURL: URL
+  currentLocation: URL
+  currentBaseURL: string
+}
+
+/**
+ * Choose a URL to navigate to when the user chooses a version in the version
+ * selector.
+ *
+ * The parameters in `params` are named as follows, in order to make it clearer
+ * which parameter means what when invoking the function:
+ *
+ *  - selectedVersionSitemap: Sitemap - as obtained by fetchSitemap from `${selectedVersionBaseURL}/sitemap.xml`
+ *
+ *  - selectedVersionBaseURL: URL - usually `${currentBaseURL}/../selectedVersion`
+ *
+ *  - currentLocation: URL - current web browser location
+ *
+ *  - currentBaseURL: string - as obtained from `config.base`
+ *
+ * @param params - arguments with the meanings explained above.
+ * @returns the URL to navigate to or null if we can't be sure that the
+ * corresponding page to the current page exists in the selected version
+ */
+export function selectedVersionCorrespondingURL(
+  params: CorrespondingURLParams
+): URL | undefined {
+  const {selectedVersionSitemap,
+    selectedVersionBaseURL,
+    currentLocation,
+    currentBaseURL} = params
+  const current_path = safeURLParse(currentBaseURL)?.pathname
+  if (current_path === undefined) {
+    return
+  }
+  const currentRelativePath = stripPrefix(currentLocation.pathname, current_path)
+  if (currentRelativePath === undefined) {
+    return
+  }
+  const sitemapCommonPrefix = shortestCommonPrefix(selectedVersionSitemap.keys())
+  if (!selectedVersionSitemap.has(sitemapCommonPrefix)) {
+    // We could also check that `commonSitemapPrefix` ends in the canonical version,
+    // similarly to https://github.com/squidfunk/mkdocs-material/pull/7227. However,
+    // I don't believe that Mike/MkDocs ever generate sitemaps where it would matter
+    return
+  }
+
+  const potentialSitemapURL = safeURLParse(currentRelativePath, sitemapCommonPrefix)
+  if (!potentialSitemapURL || !selectedVersionSitemap.has(potentialSitemapURL.href)) {
+    return
+  }
+
+  const result = safeURLParse(currentRelativePath, selectedVersionBaseURL)
+  if (!result) {
+    return
+  }
+  result.hash = currentLocation.hash
+  result.search = currentLocation.search
+  return result
+}
+
+/**
+ * A version of `new URL` that never throws. A polyfill for URL.parse() which is
+ * not yet ubuquitous.
+ *
+ * @param url - passed to `new URL` constructor
+ * @param base - passed to `new URL` constructor
+ *
+ * @returns `new URL(url, base)` or undefined if the URL is invalid.
+ */
+function safeURLParse(url: string|URL, base?: string|URL): URL | undefined {
+  try {
+    return new URL(url, base)
+  } catch {
+    return
+  }
+}
+
+// Basic string manipulation
+
+/** Strip a given prefix from a function
+ *
+ * @param s - string
+ * @param prefix - prefix to strip
+ *
+ * @returns either the string with the prefix stripped or undefined if the
+ * string did not begin with the prefix.
+ */
+export function stripPrefix(s: string, prefix: string): string | undefined {
+  if (s.startsWith(prefix)) {
+    return s.slice(prefix.length)
+  }
+  return undefined
+}
+
+/** Find the length of the longest common prefix of two strings
+ *
+ * @param s1 - first string
+ * @param s2 - second string
+ *
+ * @returns - the length of the longest common prefix of the two strings.
+ */
+function commonPrefixLen(s1: string, s2: string): number {
+  const max = Math.min(s1.length, s2.length)
+  let result
+  for (result = 0; result < max; ++result) {
+    if (s1[result] !== s2[result]) {
+      break
+    }
+  }
+  return result
+}
+
+/** Find the longest common prefix of any number of strings
+ *
+ * @param strs - an iterable of strings
+ *
+ * @returns the longest common prefix of all the strings
+ */
+export function shortestCommonPrefix(strs: Iterable<string>): string {
+  let result  // Undefined if no iterations happened
+  for (const s of strs) {
+    if (result === undefined) {
+      result = s
+    } else {
+      result = result.slice(0, commonPrefixLen(result, s))
+    }
+  }
+  return result ?? ""
+}
diff --git a/src/templates/assets/javascripts/integrations/version/index.ts b/src/templates/assets/javascripts/integrations/version/index.ts
index 50de29ad461..d6c68d278da 100644
--- a/src/templates/assets/javascripts/integrations/version/index.ts
+++ b/src/templates/assets/javascripts/integrations/version/index.ts
@@ -48,6 +48,8 @@ import {
 
 import { fetchSitemap } from "../sitemap"
 
+import { selectedVersionCorrespondingURL } from "./findurl"
+
 /* ----------------------------------------------------------------------------
  * Helper types
  * ------------------------------------------------------------------------- */
@@ -122,22 +124,23 @@ export function setupVersionSelector(
                     return EMPTY
                 }
                 ev.preventDefault()
-                return of(url)
+                return of(new URL(url))
               }
             }
             return EMPTY
           }),
-          switchMap(url => {
-            return fetchSitemap(new URL(url))
-              .pipe(
-                map(sitemap => {
-                  const location = getLocation()
-                  const path = location.href.replace(config.base, url)
-                  return sitemap.has(path.split("#")[0])
-                    ? new URL(path)
-                    : new URL(url)
-                })
-              )
+          switchMap(selectedVersionBaseURL => {
+            return fetchSitemap(selectedVersionBaseURL).pipe(
+              map(
+                sitemap =>
+                  selectedVersionCorrespondingURL({
+                    selectedVersionSitemap: sitemap,
+                    selectedVersionBaseURL,
+                    currentLocation: getLocation(),
+                    currentBaseURL: config.base
+                  }) ?? selectedVersionBaseURL,
+              ),
+            )
           })
         )
       )