The recently developed notion of TCP-compatibility has led to a number of proposals for alternative congestion control algorithms whose long-term throughput as a function of a steady-state loss rate is similar to that of TCP. Motivated by the needs of some streaming and multicast applications, these algorithms seem poised to take the current TCP-dominated Internet to an Internet where many congestion control algorithms co-exist. An important characteristic of these alternative algorithms is that they are slowly-responsive, refraining from reacting as drastically as TCP to a single packet loss.However, the TCP-compatibility criteria explored so far in the literature considers only the static condition of a fixed loss rate. This paper investigates the behavior of slowly-responsive, TCP-compatible congestion control algorithms under more realistic dynamic network conditions, addressing the fundamental question of whether these algorithms are safe to deploy in the public Internet. We study persistent loss rates, long- and short-term fairness properties, bottleneck link utilization, and smoothness of transmission rates.