Since PMPro doesn't set the expiration date to allow for recurring billing, set cancellation date to match next payment date.

pmpro-cancel-after-subscription.php

<?php // do not include in customizations plugin
/**
 *
 * Change cancellation date to match next payment
 *
 * Add to PMPro Customizations plugin
 */
add_action( 'pmpro_after_change_membership_level', 'pmpro_cancel_at_end_of_period', 10, 2 );
add_filter( 'pmpro_email_body', 'pmpro_email_body_for_cancel_msg', 10, 2 );

	/**
	 * Change cancellation to set expiration date for next payment instead of cancelling immediately.
	 * Assumes monthly membership levels.
	 * Assumes orders are generated for each payment (i.e. your webhooks/etc are setup correctly).
	 *
	 * @param (int) $level_id -- The membership levelId we're changing
	 * @param (int) $user_id -- The member we're changing levels for.
	 */
function pmpro_cancel_at_end_of_period( $level_id, $user_id ) {

	global $pmpro_pages, $wpdb;

	// Make sure we're on the 'cancel' page (for PMPro).
	if ( is_page( $pmpro_pages['cancel'] ) ) {

		/*
		*   okay, let's give the user his old level back with an expiration based on his subscription date
		*/

		/* Fetch the last order */
		$order = new MemberOrder();
		$order->getLastMemberOrder( $user_id, 'cancelled' );

		// Fetch the levelId representing their most recent (old) membership level.
		$level = $wpdb->get_row(
			$wpdb->prepare(
				"
				SELECT *
				FROM {$wpdb->pmpro_memberships_users}
				WHERE membership_id = %d AND user_id = %d
				ORDER BY id DESC
				LIMIT 1
				",
				$order->membership_id,
				$user_id
			)
		);

		// Not allowed to do this unless there's both a levelId and an order located in the system
		// This is probably wrong since $order rarely (never?) is empty.
		if ( ( ! isset( $order->user_id ) ) || empty( $level ) ) {
			return false;
		}

		if ( empty( $level->cylcle_number ) ) {
			return false;
		}

		// Calculate the "last payment date" (using "today").
		$lastdate = date( 'Y-m-d', $order->timestamp );

		// Calculating the "next scheduled date"
		$nextdate = $wpdb->get_var(
			$wpdb->prepare(
				"
				SELECT UNIX_TIMESTAMP(%s + INTERVAL %d {$level->cycle_period})
				",
				$lastdate,
				$level->cycle_number
			)
		);

		// if the date in the future?
		if ( $nextdate - current_time( 'timestamp' ) > 0 ) {

			// Set their level to the one they're currently at, but leave the expiration date set to their next billing date.
			$sql = $wpdb->prepare(
				"
				SELECT *
				FROM $wpdb->pmpro_memberships_users
				WHERE ( membership_id = %d ) AND ( user_id = %d )
				ORDER BY id
				DESC LIMIT 1
				",
				$order->membership_id,
				$user_id
			);

			$old_level = $wpdb->get_row( $sql, ARRAY_A );

			$old_level['enddate'] = date( 'Y-m-d H:i:s', $nextdate );

			// disable this hook so we don't loop
			remove_action( 'pmpro_after_change_membership_level', 'pmpro_cancel_at_end_of_period', 10, 2 );
			remove_action( 'pmpro_after_change_membership_level', 'pmpro_membership_level_config', 10, 2 );
			remove_action( 'pmpro_after_change_membership_level', 'pmpro_clear_upcoming_appointments', 11, 2 );
			remove_action( 'pmpro_after_change_membership_level', 'pmpro_after_change_membership_level', 10, 2 );

			// change level
			pmpro_changeMembershipLevel( $old_level, $user_id );

			// add the action back just in case
			add_action( 'pmpro_after_change_membership_level', 'pmpro_cancel_at_end_of_period', 10, 2 );
			add_action( 'pmpro_after_change_membership_level', 'pmpro_membership_level_config', 10, 2 );
			add_action( 'pmpro_after_change_membership_level', 'pmpro_clear_upcoming_appointments', 11, 2 );
			add_action( 'pmpro_after_change_membership_level', 'pmpro_after_change_membership_level', 10, 2 );

			// change message shown on cancel page
			add_filter( 'gettext', 'pmpro_gettext_cancel_text', 10, 3 );
		}
	}
}

	/**
	 * @param $translated_text -- Text to replace/update
	 * @param $text -- The text
	 * @param $domain
	 *
	 * @return string
	 */
function pmpro_gettext_cancel_text( $translated_text, $text, $domain ) {

	if ( ( $domain == 'pmpro' ) &&
		( $text == 'Your membership has been cancelled.' ) ) {

		global $current_user;

		$translated_text = 'Your membership has been cancelled. Your access to schedule and join new sessions will expire on ' .
		date_i18n( get_option( 'date_format' ), pmpro_next_payment( $current_user->ID, 'cancelled' ) ) .
		', but you may schedule new sessions until then. Once your membership expires, any scheduled sessions ' .
		'will be removed automatically.';
	}

	return $translated_text;
}

	/**
	 * If the user cancels, send a message about when the cancellation will actually occur.
	 *
	 * @param (string) $body - The content of the message (we'll append the cancellation date to it)
	 * @param (object) $email -- The email object.
	 *
	 * @return string -- The new body of the email message.
	 */
function pmpro_email_body_for_cancel_msg( $body, $email ) {

	if ( $email->template == 'cancel' ) {

		global $wpdb;

		$user_id = $wpdb->get_var(
			"
				SELECT ID
				FROM $wpdb->users
				WHERE user_email = '" . esc_sql( $email->email ) . "' LIMIT 1"
		);

		if ( ! empty( $user_id ) ) {

			$expiration_date = pmpro_next_payment( $user_id );

			// Only update the body of the message if the date is in the future.
			if ( $expiration_date - current_time( 'timestamp' ) > 0 ) {

				$body .= '<p>Your access will expire on ' . date( get_option( 'date_format' ), $expiration_date ) . '.</p>';
			}
		}
	}

	return $body;
}