The paper is concerned with the Hölder-continuous dependence on data that results when certain ill-posed continuation problems in partial differential equations are stabilized by prescribed bounds. The author highlights the fact that, due to the Hölder exponent decaying to zero as the continuation boundary is approached, there is a resulting growth in errors originating from imperfect data and consequently continuation into a particular region of interest is often not possible. However, for solutions satisfying the slow evolution from the continuation boundary (SECB) constraint previously introduced by the author (see [SIAM J. Numer. Anal. 31, 1535-1557 (1994; Zbl 0812.35155)], [SIAM J. Math. Anal. 28, No. 3, 656-668 (1997; Zbl 0870.35119)]), stronger stability estimates can be obtained and continuation becomes feasible. This constraint has been shown to be effective in controlling the growth of noise in certain image deblurring problems, in which a key role is played by the backwards in time continuation in diffusion equations involving fractional Laplacians. In the current paper, the author extends his earlier work on the SECB constraint to a wider range of problems, including analytic continuation on the unit disc, the time-reversed Navier-Stokes equations, time-reversed parabolic equations in \(L^p\) spaces, and a class of nonparabolic equations, involving nonlocal partial differential operators, that are obtained by subordination in well-posed abstract Cauchy problems.