There are single-name equities and broad-based indices.
1. Single-name equities.
Here the dollar dividends are discrete. I wouldn't use a 'rate'. Treating discrete dollar dividends sensibly under Black-Scholes is both tricky and contentious: there are various approaches. The one I advocate is given in Ch. 9 ('Back to Basics: an update on the discrete dividend problem') in my book "Option Valuation under Stochastic Volatility II". Before reading that chapter, you might also want to take a look at the 2003 article from Wilmott magazine, similarly titled "Back to basics: ..." by Haug, Haug, and Lewis. (available in various places).
It's possible your implied dividends are not behaving sensibly because you are using an older (improper, IMO) escrowed dividend model. The problems with escrowed dividends are discussed in both of my sources for you.
2. Broad-based indices.
Here, a time-varying dividend rate makes sense to extract. To do so, I recommend extracting an inferred dividend rate by the method of the "VIX white paper". The method amounts to fixing an interest rate and then extracting an option-implied forward price for the underlying, with the maturity of the forward contract identical to the maturity of the options. Of course, if it is for an important purpose, you can always try to reconcile that rate with direct day-by-day dividend projections.