We study bifurcation phenomena in natural families of rational, (transcendental) entire or meromorphic functions of finite type ${f_lambda := varphi_lambda circ f_{lambda_0} circ psi^{-1}_lambda}_{lambdain M}$, where $M$ is a complex connected manifold, $lambda_0in M$, $f_{lambda_0}$ is a meromorphic map and $varphi_lambda$ and $psi_lambda$ are families of quasiconformal homeomorphisms depending holomorphically on $lambda$ and with $psi_lambda(infty)=infty$. There are fundamental differences compared to the rational or entire setting due to the presence of poles and therefore of parameters for which singular values are eventually mapped to infinity (singular parameters). Under mild geometric conditions we show that singular (asymptotic) parameters are the endpoint of a curve of parameters for which an attracting cycle progressively exits de domain, while its multiplier tends to zero. This proves the main conjecture by Fagella and Keen (asymptotic parameters are virtual centers) in a very general setting. Other results in the paper show the connections between cycles exiting the domain, singular parameters, activity of singular orbits and $J$-unstability, converging to a theorem in the spirit of the celebrated result by Ma~{n}e-Sad-Sullivan and Lyubich.
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