Are trends in crossplots of compositional data from different analyses valid?
The question is not exactly clear. Are you asking about the different methods used to analyse the rock, or are you asking about using closed (aka compositional) data in a bivariate plot?

Using different methods

This is just fine, as long as both methods are accurate. For instance, you can analyse a mineral for major elements (e.g. TiO2) using an electron probe, and then use LA-ICP-MS for trace elements (e.g. Nb). If both methods are properly standardised and all possible interferences are taken care of there is no reason why the analyses should not be used together. If the analysis was not performed properly, I wouldn't trust even as single one, regardless of whether it is used in a database together with data acquired using a different method.

Closed data

Closed data means that it has to sum to 1 or 100%, and this can cause correlations that don't necessarily mean anything. A simple example is an MgO vs FeO plot in olivine. Of course there would be a negative correlation! This type of plot would be obvious and wouldn't tell us a thing. However, in more complicated cases, this becomes less and more of a problem. It mostly depends on a case by case basis. Note that if you're comparing a major element (TiO2) with a trace element (Nb), there is less of a problem. The trace element is not one of the components that hardly has any implications for the close problem. Whether you have 10, 20 or 100 ppm Nb will not change anything for the 5% of TiO2 you got in your rock.

Classic statistics done on compositional data is almost always wrong. In the past 20 years or so, a new statistical field called "compositional data analysis" has emerged to solve some of the problems arising from the closure problem. In short, it means that you have to transform all of your data to log-ratio form, do your stats, and transform it back. This can't solve all of the problems, however.

Now, the question is - do we care that it's wrong? In the vast majority of papers, there is hardly any statistics done of compositional data, and many correlations and trends are eyeballed and drawn by hand. Often, the hassle of doing statistics the correct way does not actually contribute everything. I've heard senior petrologists claim more than once that "if it looks like a line, it is a line. If it doesn't, then it is not a line. I don't need the R2 and p-values for that".