ABSTRACT We explore deep rest-frame UV to FIR data in the COSMOS field to measure the individual spectral energy distributions (SED) of the ∼4000 SC4K (Sobral et al.) Lyman α (Ly α) emitters (LAEs) at z ∼ 2–6. We find typical stellar masses of 109.3 ± 0.6 M⊙ and star formation rates (SFR) of SFR$_{\rm SED}=4.4^{+10.5}_{-2.4}$ M⊙ yr−1 and SFR$_{\rm Ly\,α }=5.9^{+6.3}_{-2.6}$ M⊙ yr−1, combined with very blue UV slopes of $β =-2.1^{+0.5}_{-0.4}$, but with significant variations within the population. MUV and β are correlated in a similar way to UV-selected sources, but LAEs are consistently bluer. This suggests that LAEs are the youngest and/or most dust-poor subset of the UV-selected population. We also study the Ly α rest-frame equivalent width (EW0) and find 45 ‘extreme’ LAEs with EW0 > 240 Å (3σ), implying a low number density of (7 ± 1) × 10−7 Mpc−3. Overall, we measure little to no evolution of the Ly α EW0 and scale length parameter (w0), which are consistently high (EW$_0=140^{+280}_{-70}$ Å, $w_0=129^{+11}_{-11}$ Å) from z ∼ 6 to z ∼ 2 and below. However, w0 is anticorrelated with MUV and stellar mass. Our results imply that sources selected as LAEs have a high Ly α escape fraction (fesc,Ly α) irrespective of cosmic time, but fesc,Ly α is still higher for UV-fainter and lower mass LAEs. The least massive LAEs (<109.5 M⊙) are typically located above the star formation ‘main sequence’ (MS), but the offset from the MS decreases towards z ∼ 6 and towards 1010 M⊙. Our results imply a lack of evolution in the properties of LAEs across time and reveals the increasing overlap in properties of LAEs and UV-continuum selected galaxies as typical star-forming galaxies at high redshift effectively become LAEs.