Regression of first differenced log-transformed model I am cosidering to transform a regression equation applying logarithms to the dependent and some of the independent variables (the ones I am actually interested in while leaving unchanged the others that I would like to use just as controls). This in order to interpret the coefficients of the transformed variables, say b, as "1% change in explanatory variable induces b% change in the dependent variable". Then I am thinking to first-differencing transform the model, that is, to introduce for both dependent and independent variables (again just the ones log-transformed of interest) the first lagged values (the values in the previous period) and the taking the differences between current and previous values as variables of the regression to be estimated. Wuold there be any issue in following this procedure?
What you describe is very standard in academic econometric practice. It essentially answers what is the elasticity of Y with respect to X identified from changes in X and Y. 

Here are a few problems that can be introduced by this process


First-differencing is very similar to $Y_t = \alpha + \beta_0 * Y_{t-1} + \beta_1 * X_{t} + \beta_2 * X_{t-1} + \epsilon$ with the additional restrictions that $\beta_0 = 1$ and $\beta_1 = -\beta_2$
First-differencing can introduce serial correlation in standard errors