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Analysis of all-sky Planck submillimetre observations and the IRAS 100um data has led to the detection of a population of Galactic cold clumps. The clumps can be used to study star formation and dust properties in a wide range of Galactic environments. Our aim is to measure dust spectral energy distribution (SED) variations as a function of the spatial scale and the wavelength. We examine the SEDs at large scales using IRAS, Planck, and Herschel data. At smaller scales, we compare with JCMT/SCUBA-2 850um maps with Herschel data that are filtered using the SCUBA-2 pipeline. Clumps are extracted using the Fellwalker method and their spectra are modelled as modified blackbody functions. According to IRAS and Planck data, most fields have dust colour temperatures T_C ~ 14-18K and opacity spectral index values of beta=1.5-1.9. The clumps/cores identified in SCUBA-2 maps have T~ 13K and similar beta values. There are some indications of the dust emission spectrum becoming flatter at wavelengths longer than 500um. In fits involving Planck data, the significance is limited by the uncertainty of the corrections for CO line contamination. The fits to the SPIRE data give a median beta value slightly above 1.8. In the joint SPIRE and SCUBA-2 850um fits the value decreases to beta ~1.6. Most of the observed T-beta anticorrelation can be explained by noise. The typical submillimetre opacity spectral index beta of cold clumps is found to be ~1.7. This is above the values of diffuse clouds but lower than in some previous studies of dense clumps. There is only tentative evidence of T-beta anticorrelation and beta decreasing at millimetre wavelengths.