How splicing factors are recruited to nascent transcripts in the nucleus in order to assemble spliceosomes on newly synthesised pre-mRNAs is unknown. To address this question, we compared the intranuclear trafficking kinetics of small nuclear ribonucleoprotein particles (snRNP) and non-snRNP proteins in the presence and absence of splicing activity. Photobleaching experiments clearly show that spliceosomal proteins move continuously throughout the entire nucleus independently of ongoing transcription or splicing. Using quantitative experimental data, a mathematical model was applied for spliceosome assembly and recycling in the nucleus. The model assumes that splicing proteins move by Brownian diffusion and interact stochastically with binding sites located at different subnuclear compartments. Inhibition of splicing, which reduces the number of pre-mRNA binding sites available for spliceosome assembly, was modeled as a decrease in the on-rate binding constant in the nucleoplasm. Simulation of microscopy experiments before and after splicing inhibition yielded results consistent with the experimental observations. Taken together, our data argue against the view that spliceosomal components are stored in nuclear speckles until a signal triggers their recruitment to nascent transcripts. Rather, the results suggest that splicing proteins are constantly diffusing throughout the entire nucleus and collide randomly and transiently with pre-mRNAs.