diff --git a/bin/update-readmes.js b/bin/update-readmes.js
index 1aca8699f9c07b..63d041e7ab5098 100755
--- a/bin/update-readmes.js
+++ b/bin/update-readmes.js
@@ -1,8 +1,7 @@
 #!/usr/bin/env node
 
 const { join } = require( 'path' );
-const { promisify } = require( 'util' );
-const spawn = promisify( require( 'child_process' ).spawn );
+const spawnSync = require( 'child_process' ).spawnSync;
 
 const packages = [
 	'a11y',
@@ -39,15 +38,17 @@ const packages = [
 	'wordcount',
 ];
 
-Promise.all( packages.map( ( entry ) => {
+packages.forEach( ( entry ) => {
 	if ( ! Array.isArray( entry ) ) {
 		entry = [ entry, { 'Autogenerated API docs': 'src/index.js' } ];
 	}
 
 	const [ packageName, targets ] = entry;
 
-	return Promise.all( Object.entries( targets ).map( async ( [ token, path ] ) => {
-		const { status, stderr } = await spawn(
+	Object.entries( targets ).forEach( ( [ token, path ] ) => {
+		// Each target operates over the same file, so it needs to be processed synchronously,
+		// as to make sure the processes don't overwrite each other.
+		const { status, stderr } = spawnSync(
 			join( __dirname, '..', 'node_modules', '.bin', 'docgen' ),
 			[
 				join( 'packages', packageName, path ),
@@ -60,10 +61,8 @@ Promise.all( packages.map( ( entry ) => {
 		);
 
 		if ( status !== 0 ) {
-			throw stderr.toString();
+			process.stderr.write( `${ packageName } ${ stderr.toString() }\n` );
+			process.exit( 1 );
 		}
-	} ) );
-} ) ).catch( ( error ) => {
-	process.stderr.write( `${ error }\n` );
-	process.exit( 1 );
+	} );
 } );
diff --git a/package-lock.json b/package-lock.json
index 7211447a4e733c..5f830dff76a576 100644
--- a/package-lock.json
+++ b/package-lock.json
@@ -3673,20 +3673,12 @@
 			}
 		},
 		"acorn-jsx": {
-			"version": "3.0.1",
-			"resolved": "https://registry.npmjs.org/acorn-jsx/-/acorn-jsx-3.0.1.tgz",
-			"integrity": "sha1-r9+UiPsezvyDSPb7IvRk4ypYs2s=",
+			"version": "4.1.1",
+			"resolved": "https://registry.npmjs.org/acorn-jsx/-/acorn-jsx-4.1.1.tgz",
+			"integrity": "sha512-JY+iV6r+cO21KtntVvFkD+iqjtdpRUpGqKWgfkCdZq1R+kbreEl8EcdcJR4SmiIgsIQT33s6QzheQ9a275Q8xw==",
 			"dev": true,
 			"requires": {
-				"acorn": "^3.0.4"
-			},
-			"dependencies": {
-				"acorn": {
-					"version": "3.3.0",
-					"resolved": "https://registry.npmjs.org/acorn/-/acorn-3.3.0.tgz",
-					"integrity": "sha1-ReN/s56No/JbruP/U2niu18iAXo=",
-					"dev": true
-				}
+				"acorn": "^5.0.3"
 			}
 		},
 		"agent-base": {
@@ -8770,13 +8762,13 @@
 			"dev": true
 		},
 		"espree": {
-			"version": "3.5.4",
-			"resolved": "https://registry.npmjs.org/espree/-/espree-3.5.4.tgz",
-			"integrity": "sha512-yAcIQxtmMiB/jL32dzEp2enBeidsB7xWPLNiw3IIkpVds1P+h7qF9YwJq1yUNzp2OKXgAprs4F61ih66UsoD1A==",
+			"version": "4.0.0",
+			"resolved": "https://registry.npmjs.org/espree/-/espree-4.0.0.tgz",
+			"integrity": "sha512-kapdTCt1bjmspxStVKX6huolXVV5ZfyZguY1lcfhVVZstce3bqxH9mcLzNn3/mlgW6wQ732+0fuG9v7h0ZQoKg==",
 			"dev": true,
 			"requires": {
-				"acorn": "^5.5.0",
-				"acorn-jsx": "^3.0.0"
+				"acorn": "^5.6.0",
+				"acorn-jsx": "^4.1.1"
 			}
 		},
 		"esprima": {
diff --git a/package.json b/package.json
index f2eaee420b4dfe..7986263f49cfe0 100644
--- a/package.json
+++ b/package.json
@@ -92,7 +92,7 @@
 		"doctrine": "2.1.0",
 		"enzyme": "3.9.0",
 		"eslint-plugin-jest": "21.5.0",
-		"espree": "3.5.4",
+		"espree": "4.0.0",
 		"fbjs": "0.8.17",
 		"glob": "7.1.2",
 		"husky": "0.14.3",