We show that different ways of extracting time scales from time-dependent velocity structure functions lead to different dynamic-multiscaling exponents in fluid turbulence.  These exponents are related to equal-time multiscaling exponents by different classes of bridge relations which we derive. We check this explicitly by detailed numerical simulations of the GOY shell model for fluid turbulence.  Next we apply these ideas to the problem of turbulent advection of a passive scalar. Here, our principal  results, obtained analytically and  numerically, illustrate important  principles that appear, at first sight,  to be surprising.  We find, e.g., that the dynamic exponents depend via bridge relations only on the  equal-time scaling exponents of the velocity field.  Thus, even though equal-time structure functions for the passive-scalar and passive-vector problems display multiscaling, they show simple dynamic scaling if the advecting velocity is of the Kraichnan type. Dynamic multiscaling is obtained only if the advecting velocity field is itself intermittent.