Hi Lou,
Excellent questions. I would like to add a few thoughts to help frame your approach here.
1. As mentioned by others, you definitely want to merge _all_ your samples together into a singular analysis. If you think of it this way, you want the clustering algorithm to run on a single dataset, which may (or may not) be comprised of multiple samples/groups etc. It's almost like your are blinding the algorithm to the presence of multiple samples, and then unmixing them after the analysis is complete. Depending on the software you use, it may require you to explicitly merge these together before the analysis, or it might do this 'under the hood' so check the instructions carefully.
2. Try not to think of this analysis as 'tSNE analysis' -- tools like tSNE and UMAP are helpful for visualising things with single-cell resolution, but they a) don't actually do the quantification or statistical work and b) have limitations to do with interpretations etc. The algorithms doing the heavy lifting are typically clustering or perhaps classification algorithms. This is also important for scale -- to get a reasonable statistical analysis, you want as many cells per sample as possible. Clustering algorithms like FlowSOM can take tens of millions of cells no problem, but tSNE will struggle above ~100K. Other options like Opt-SNE, FIt-SNE, or UMAP will handle larger numbers of cells, but all suffer from the same fundamental problem that they won't scale well. In this case it doesn't matter too much -- you can run the full analysis using clustering, and just visualise a subset of cells using tSNE/UMAP etc.
3. We detail such a workflow in our analysis toolkit
Spectre. It is an R package, but it requires interacting with code instead of a point-and-click interface, but it's designed to be user friendly for wet-lab scientists, so if you want to give it a try feel free. The key feature here that may be most helpful for you is that it is designed explicitly for rapid processing of very large datasets, aided by the data.table framework in R. We also detail how to replicate the design of our workflow (i.e. merge samples --> cluster --> downsample --> tSNE/UMAP/whatever) in other programs like FlowJo. You can use the same strategy we outlike in other programs too, including CytofKitLab.
https://immunedynamics.io/spectre/Check out the getting started tutorials, and then the 'simple discovery' workflow is the one to use for the kind of analysis you have described (R and FlowJo versions). The demo dataset is CNS cells from group of 8 mice (4x mock and 4x virus infected).
If you need any help, feel free to reach out -- you can reply here, or we have a discussion board
https://github.com/ImmuneDynamics/Spectre/discussions, or you can email us directly.
Good luck
Tom