Difference between scTransform/dimension reduction on merged object vs. individual

[NickOGrady]

Hello all,

I am working on a single cell project that has multiple samples from multiple participants over different timeframes. After looking through the vignettes, I'm having trouble understanding the differences from running scTransform() on a merged object of each sample (which imports as layers), versus normalizing and then running dimension reduction on each object individually.
As a general example, I have these 3 samples from the same participant as a 10x seurat object.:

xpMerge <- merge(S1, c(S2, S3))
xpMerge <- SCTransform(xpMerge, verbose = T)  
xpMerge <- RunPCA(xpMerge, verbose = FALSE)  
xpMerge <- RunUMAP(xpMerge, dims = 1:30, verbose = FALSE)  
xpMerge <- RunTSNE(xpMerge)  
xpMerge <- FindNeighbors(xpMerge, dims = 1:30, verbose = FALSE)  
xpMerge <- FindClusters(xpMerge, verbose = FALSE)   

DimPlot(xpMerge, reduction = "tsne", group.by = "seurat_clusters", split.by = "Sample",
        label = T, label.box = T) +
  labs(title = "scTransform/TSNE on merged obj")

Versus performing each of the previous steps on a sample at a time, through a loop:

xpList = merge(S1, c(S2, S3))
xpList =  SplitObject(xpList, split.by = "seuratID")

for (i in 1:length(xpList)) {
  
  xpList[[i]] <- SCTransform(xpList[[i]], verbose = T)
  xpList[[i]] <- RunPCA(xpList[[i]], verbose = FALSE)
  xpList[[i]] <- RunUMAP(xpList[[i]], dims = 1:30, verbose = FALSE)
  xpList[[i]] <- RunTSNE(xpList[[i]])
  xpList[[i]] <- FindNeighbors(xpList[[i]], dims = 1:30, verbose = FALSE)
  xpList[[i]] <- FindClusters(xpList[[i]], verbose = FALSE)
  
}

# more code to plot the bottom row below

I noticed the resulting TSNE plots are different, but I don't know the source of the differences.

The differences are more subtle when only considering these 3 samples from the same pariticpant, but become more obvious when more samples and participants are involved. Any help in understanding these mechanisms would be greatly appreciated.
I am using Seurat package version 5.1.0.

Thank you.