The Internet is based on best effort communication, i.e. it tries to deliver packets but does not provide any guarantees. A transport protocol can make use of this best effort service to provide a suitable service to its applications. Also, its congestion control is responsible for a fair distribution of the resources within the Internet. However, background data transfer applications (like file sharing or update fetching) do not require "best effort"; they in fact could use a "lower-than-best-effort" service to leave resources to more important applications if needed. For this purpose, the Low Extra Delay Background Transport (LEDBAT) algorithm has been standardized by the IETF. Nowadays, multi-homing is becoming increasingly common in modern networks and several approaches to exploit this feature (e.g. CMT-SCTP, MPTCP) have evolved that are able to combine resources of multiple paths. For background traffic oriented algorithms like LEDBAT, this feature could be of great use, too, i.e. by increasing the overall bandwidth while shifting the transmission away from paths which are used by other flows. This could be particularly useful for non-critical bulk transfers in data centers. In this paper, we introduce our approach LEDBAT for Multi-Path - denoted as LEDBAT-MP - and analyse its performance by simulations. With this paper, we want to highlight some generic design questions and start a discussion on how a solid universal background multi-path congestion control strategy should behave.