With the deployment of more and more resilience-critical Internet applications, there is a rising demand for multi-homed network sites. This leads to the desire for simultaneously utilising all available access paths to improve application data throughput. This is commonly known as Concurrent Multipath Transfer (CMT); approaches for several Transport Layer protocols have been proposed. Combined with Resource Pooling (RP), CMT can also fairly coexist with concurrent non-CMT flows. Current approaches focus on symmetric paths (i.e. similar bandwidth, delay and error rate). However, asymmetric paths are much more likely - particularly for realistic Internet setups - and efficient CMT usage on such paths is therefore crucial. In this paper, we first show the challenges of plain as well as RP-aware CMT data transport over asymmetric paths. After that, we introduce mechanisms for efficient transport over such paths. Finally, we analyse the performance of our approaches by using simulations.