On-the-fly Garbage Collectors (GCs) are the state-of-the-art concurrent GC algorithms today. Everything is done concurrently, but phases are separated by blocking handshakes. Hence, progress relies on the scheduler to let application threads (mutators) run into GC checkpoints to reply to the handshakes. For a non-blocking GC, these blocking handshakes need to be addressed. Therefore, we propose a new non-blocking handshake to replace previous blocking handshakes. It guarantees schedulingindependent operation level progress without blocking. It is scheduling independent but requires some other OS support. It allows bounded waiting for threads that are currently running on a processor, regardless of threads that are not running on a processor. We discuss this non-blocking handshake in two GC algorithms for stack scanning and copying objects. They pave way for a future completely non-blocking GC by solving hard open theory problems when OS support is permitted. The GC algorithms were integrated to the G1 GC of OpenJDK for Java. GC pause times were reduced to 12.5% compared to the original G1 on average in DaCapo. For a memory intense benchmark, latencies were reduced from 174 ms to 0.67 ms for the 99.99% percentile. The improved latency comes at a cost of 15% lower throughput.