I read in a paper that the discrete wavelet transform (DWT) has two disadvantages The first one is the shift variance property due to the downsampling process. Could you please help me understanding why downsampling leads to shift variance? The second disadvantage is the lack of directional selectivity. Why does DWT lacks directional selectivity?

A $$2$$-channel stage of a wavelet transform, combines two filters in parallel, followed by a down-sampling by two. The later is the cause for shift-invariance, as the filters are time-invariant. Signals $$x_0[n] = \{\ldots,0,1,0,1,0,1,\ldots\}$$ and $$x_1[n] = \{\ldots,1,0,1,0,1,0,\ldots\}$$ which are shifted by only one sample, yield respectively $$y_0[n] = \{\ldots,0,0,0,0,0,0,\ldots\}$$ and $$y_1[n] = \{\ldots,1,1,1,1,1,1,\ldots\}$$
Answers to down-sampling a thus not shift-invariant. This happens even with low-pass or high-pass filters. In an $$L$$-level wavelet decomposition, the overall wavelet filter bank is invariant to multiples of $$2^L$$ shift, not the intermediate one.