This paper considers dynamic term structure models like the ones appearing in portfolio credit risk modelling or life insurance. We study general forward rate curves driven by infinitely many Brownian motions and an integer-valued random measure, generalizing existing approaches in the literature. A precise characterization of absence of arbitrage in such markets is given in terms of a suitable criterion for no asymptotic free lunch (NAFL). From this, we obtain drift conditions which are equivalent to NAFL. In a suitable special case we are able to derive existence results. For applications, models possessing a certain monotonicity are favorable and we study general conditions which guarantee this. The setup is illustrated with some examples.